(navigation image)
Home American Libraries | Canadian Libraries | Universal Library | Community Texts | Project Gutenberg | Children's Library | Biodiversity Heritage Library | Additional Collections
Search: Advanced Search
Anonymous User (login or join us)
See other formats

Full text of "Introduction to Physical Chemistry"

f» ^±a^jg^r:^sS^L J* £7/ 




i^ ^s^^&" ^^ 5j 

;^^=rvj ^^3^ 


5? ^^^^"f^Sfe^^ rtr- 




' ' 'AV^ 





g^ a w , S 5 a 






















All rights reserved 




This book makes no pretension to give a complete or even systematic 
survey of Physical Chemistry ; its main object is to be explanatory. 
I have found, in the course of ten years 7 experience in teaching the 
subject, that the average student derives little real benefit from reading 
the larger works which have hitherto been at his disposal, owing chiefly 
to his inability to effect a connection between the ordinary chemical 
knowledge he possesses and the new material placed before him. He 
keeps his everyday chemistry and his physical chemistry strictly apart, 
with the result that instead of obtaining any help from the new 
discipline in the comprehension of his systematic or practical work, 
he merely finds himself cumbered with an additional burthen on the 
memory, which is to all intents and purposes utterly useless. This 
state of affairs I have endeavoured to remedy in the present volume 
by selecting certain chapters of Physical Chemistry and treating the 
subjects contained in them at some length, with a constant view to 
their practical application. In choice of subjects and mode of treatment 
I have been guided by my own teaching experience. I have striven 
to smooth, as far as may be, the difficulties that beset the student's 
path, and to point out where the hidden pitfalls lie. If I have been 
successful in my object, the student, after a careful perusal of this 
introductory text-book, should be in a position to profit by the study 
of the larger systematic works of Ostwald, Nernst, and van 't Hoff. 

As I have assumed that the student who uses this book has already 
taken ordinary courses in chemistry and physics, I have devoted little 
or no space to the explanation of terms or elementary notions which 



are adequately treated in the text-books on those subjects. I have 
throughout avoided the use of any but the most elementary mathematics, 
the only portion of the book requiring a rudimentary knowledge of the 
calculus being the last chapter, which contains the thermodynamical 
proofs of greatest value to the chemist. 

Since it is of the utmost importance that even beginners in physical 
chemistry should become acquainted at first hand with original work 
on the subject, I have given a few references to papers generally 
accessible to English-speaking students. 


August 1899. 




Units and Standards of Measurement ... 1 


The Atomic Theory and Atomic Weights ... 8 


Chemical Equations . . . . . .22 


The Simple Gas Laws . . - . . ,27 

Specific Heats . . . . • .30 


The Periodic Law ...... 38 




Solubility ....... 50 


Fusion and Solidification ..... 60 


Vapoeisation and Condensation . . . .73 


The Kinetic Theory and Van Der Waals's Equation . 84 


The Phase Rule ...... 97 



Thermochemical Change . . . . .117 


Variation op Physical Properties in Homologous Series . 127 


Relation of Physical Properties to Composition and Constitution 1 36 





The Properties of Dissolved Substances . 148 


Osmotic Pressure and the Gas Laws for Dilute Solutions 158 


Deductions from the Gas Laws for Dilute Solutions . 169 


Methods of Molecular Weight Determination . . 176 


Molecular Complexity . . . . . .193 


Electrolytes and Electrolysis . . . .201 


Electrolytic Dissociation . . . . .217 


Balanced Actions . . . . . .234 




Rate of Chemical Transformation . . .254 


Relative Strengths of Acids and Bases . .266 


Equilibrium between Electrolytes . . .283 


Applications of the Dissociation Theory . . . 296 


Thermodynamical Proofs . . . .311 


INDEX ........ 333 



To express the magnitude of anything, we use in general a number 
and a name. Thus we speak of a length of 3 feet, a temperature 
difference of 18 degrees, and so forth. The name is the name of the 
unit in terms of which the magnitude is measured, and the number 
gives the number of times this unit is contained in the given magni- 
tude. The selection of the unit in each case is arbitrary, and regulated 
solely by our convenience. In different countries different units of 
"length are in vogue, and even in the same country it is found con- 
venient to adopt sometimes one unit, sometimes another. Lengths, 
for example, when very great are expressed in miles ; when small, in 
inches : and we also find in use the foot and yard as units for inter- 
mediate lengths. Such units as these British measures of length 
were fixed by custom and convention, and for the purposes of every- 
day life are convenient enough. When we come, however, to the 
discussion of scientific problems, we find that they are unsuitable and 
inconvenient, leading to clumsy calculations the greater part of which 
could be dispensed with if the units of the various magnitudes were 
properly selected. There is, for instance, no simple relation between 
any of the British units of length and the usual British standard of 
capacity — the gallon. It is true that the cubic inch and cubic foot 
are sometimes used as measures of capacity, but not generally for 
liquids. Now lengths, volumes, and weights often enter in such a 
way into scientific calculations that the existence of simple relations 
between the units of these magnitudes enables us to perform a 
calculation mentally which would necessitate a tedious arithmetical 
operation were the units not thus simply related. 

The first requirement of a convenient system of measurement is 
that all multiples and subdivisions of the unit chosen should be 
decimal, in order to be in harmony with our decimal system of 
numeration. For scientific purposes the decimal principle of measure- 
ment is uniformly accepted, save in circular measure and in the 


2 A \ ^i3^cbtJtSTj0J( -TO PHYSICAL CHEMISTEY chap. 

measurement of time, where the ancient sexagesimal system (of 
counting by sixties) still in part prevails. The unit of length is 
subdivided into tenths, hundredths, and thousandths if we wish to 
use the smaller derived units \ if we desire a larger derived unit we 
take it ten, a hundred, or a thousand times greater than the funda- 
mental unit. The choice of this fundamental unit is, as has been 
already stated, quite arbitrary. At the time of the French Eevolution, 
when a decimal system was first adopted in a thoroughgoing manner, 
the unit of length, the metre, was selected because it was of a 
convenient length for practical measurement, and in particular because 
it was supposed to have a natural relation to the size of the earth, ten 
million metres measuring exactly the quadrant of a circle through the 
poles. The value of the fundamental length therefore depended 
theoretically on the determination of the length of the earth's 
meridional quadrant. But the degree of accuracy with which this 
determination can be made is far inferior to that obtainable in 
comparing two short lengths (say a metre) together. If the metre 
then were strictly defined by the earth's dimensions, the fundamental 
unit of length would change with every fresh determination of the 
polar circumference. For practical purposes the metre is legally 
defined as the distance, under certain conditions, between two marks 
on a rod of platinum-iridium preserved in Paris, and the supposed 
exact relation to the earth's circumference has been given up. 1 
Copies of this standard have been made and distributed, and the 
relation between them and similar standards of length, such as the 
yard, have been determined with great exactness. 

For many scientific purposes, the hundredth part of the metre, 
the centimetre, is a convenient unit, and in what follows we shall 
frequently make use of it in calculations. 

The unit of mass (or weight), 2 the gram, was primarily defined 
as the mass (or weight) of 1 cubic centimetre of water at the 
temperature at which its density is greatest, viz. 4° C. Here we 
depend on the constancy of the properties of an arbitrary substance 
(pure water) to establish a relation between the units of weight and 
of cubical capacity, or volume. The original standard kilogram was 
constructed in accordance with this relation, being made equal to 
1000 grams as above defined, i.e. equal in weight to a cubic 
decimetre of water at its maximum density point. Since it is 
possible, however, to compare weights with each other with much 
greater accuracy than is attainable in the measurement of volumes, the 
exact relationship between the two units has been allowed to drop, 

1 According to recent measurement, the ten-millionth part of the earth's quadrant is 
nearly 0*2 millimetres longer than the standard metre. 

2 When a chemist uses an ordinary balance he directly compares weights, but, since 
at any one place weight is proportional to mass, he indirectly compares the masses of the 
substances on the two pans. So far, then, as measurement of quantity of material by 
means of the balance is concerned, the terms are interchangeable. 


and for exact purposes the kilogram is defined as the weight of the 
platinum standard kilogram kept in Paris. 1 For the purposes of the 
chemist the relation between the units of weight and volume as 
originally defined may be looked upon as exact, since the error is not 
greater than 0*01 per cent, a degree of accuracy to which the chemist 
attains only in exceptional circumstances. For measuring the volume 
of liquids, the litre is defined, not as a cubic decimetre, but as the 
volume occupied by a quantity of water which will balance the 
standard kilogram in vacuo at 4° C. 

The unit of time seldom enters into chemical calculations. The 
standard for the unit is derived from the length of time necessary 
for the performance of some cosmical process, for example, the time 
taken by the earth to perform a complete revolution on its axis. For 
chemical purposes, the minute, as measured by a good -going clock or 
watch, is the practical unit. 

The units of length, weight, and time being once fixed, a great 
many derived units may be fixed in their turn. The C.G.S. 
system, which takes, as the letters indicate, the centimetre, the 
gram, and the second for the fundamental units, is very frequently 
employed, especially in theoretical calculations, and we shall often 
have occasion to use it. For example, instead of expressing the 
average atmospheric pressure as that of a column of mercury 76 cm. 
high, it is expedient in many calculations to give the pressure in 
grams weight per square centimetre. The conversion may easily be 
performed as follows. Suppose the cross section of the mercury 
column to be 1 sq. cm., then if the height of the column is 76 cm., the 
total volume of mercury is 76 cc, and the pressure per square centi- 
metre is the weight of 76 cc. of mercury, viz. 1033 g. 2 The 
average pressure of the atmosphere, then, is equal to 1033 g. per 
square centimetre. 

The specific volume of a substance may be defined as the number 
of units of volume which are occupied by unit weight of the substance. 
In the above system it is therefore the number of cubic centimetres 
occupied by one gram. The specific weight, or density, of a substance 
is the number of units of weight which occupy unit volume : in the 
above system, the number of grams which occupy one cubic centimetre. 

1 To give an idea of the difficulty of accurate measurement when volumes are con- 
cerned, the following example may suffice. By Act of Parliament a gallon is equal to 
4*543458 litres, this number being derived from the weight of a cubic inch of water in 
grains, and the relation between the inch and the decimetre, a litre being supposed equal 
to a cubic decimetre. The original definition of the gallon is " the volume at 62° F. of a 
quantity of water which balances 10 brass pound weights (true in vacuo, and of specific 
gravity 8*143) in air at 62° F. and 30 inches pressure (barometer reduced to the freezing 
point) and two-thirds saturated with moisture. " From this definition, and the relation 
of the pound to the kilogram, Dittmar calculated that 1 gallon is equal to 4*54585 litres, 
a value which differs from the statutory relation by 1 part in 2000. 

2 This value is obtained by multiplying 76 by the specific gravity of mercury, viz. 


These definitions are directly derived from the units of weight and 
volume, and are independent of the properties of any substance save 
that considered. For purposes of measurement in the case of solids 
and liquids, however, it is much more convenient and accurate to com- 
pare the density or specific volume of one substance with that of an- 
other arbitrarily chosen as standard, than to effect an absolute deter- 
mination by measuring both weight and volume of one and the same 
substance. The substance usually chosen as the standard of compari- 
son is water, and that for two reasons. In the first place, water is 
easily obtained and easily purified ; in the second place, water is a 
standard substance in other respects, in particular it is the substance 
which was used to fix the relation between the units of weight and 
of volume. From this relation the density of water in our units 
is 1 (cp. p. 2), so that if we take water as our standard, and refer all 
densities to it as the unit, we have what are really relative densities, 
although if measured at 4° the numbers obtained do not differ greatly 
from the absolute densities of the substances. The actual comparison 
is made in the case of liquids by weighing the same tared vessel 
filled with water, and filled with the liquid whose density is to be 
determined. Although the actual volume is unknown, it is known 
to be the same in both cases, so that the quotient of the weight 
of liquid by the weight of water gives the relative density of the 

In order to specify with exactness the density of a substance, it is 
necessary to indicate at what temperature the determination or com- 
parison has been made. As substances change in temperature they 
invariably change in volume, so that the absolute density varies with 
the temperature. In specifying the relative density, or specific gravity, 
it is not only necessary to give the temperature at which the substance 
is weighed, but also the temperature at which the equal volume of 
water is weighed. Thus we meet with such data as the following, 
15<5 S 4 = 1*0653, which indicates that the weight of the substance at 
15*5° is 1*0653 times as much as the weight of the same volume of 
water at 4°. As a general rule, the weights are both for the same 
temperature, say 15°; or the substance is weighed at this tempera- 
ture and referred either directly or by calculation to water at 4°, its 
maximum density point. In the latter case the density given is 
numerically equal to the absolute density of the substance at the 
specified temperature. 

Quantities of matter may always be expressed in terms of the unit 
of weight, irrespective of the form in which the matter may exist. 
When we come to measure energy we find no such common unit in 
which its amount may be expressed. Each form of energy, such as 
heat, work, electrical energy, has its own special unit, but according to 
the Law of Conservation of Energy, a given amount of any one form of 
energy is under all circumstances equivalent to constant amounts of 


the other kinds of energy ; we have therefore merely to ascertain the 
relation between the different special energy units in order to express 
a given amount of energy in terms of any one of these units. Thus 
electrical energy is usually expressed in volt-coulombs for practical 
purposes, but it may easily be expressed in terms of the mechanical 
or heat units, the numbers then indicating the amounts of 
mechanical or thermal energy into which the electrical energy may 
be converted. 

The " absolute " unit of mechanical energy, the erg, is the product 
of the unit of length into the "absolute" unit of force, or dyne, i.e. the 
force necessary to impart to a mass of 1 g. in a second an acceleration 
of 1 cm. per second. For our purposes, however, it will be more 
convenient to use gravitation units, in the definition of which the 
value of the earth's gravitational attraction for bodies on its surface is 
involved. The unit of force thus defined is the weight of 1 g., and is 
equal to 981 dynes. The unit of mechanical energy is therefore the 
product of this unit into the unit of length, i.e. the gram- centimetre. 

To obtain the unit of heat and a scale of temperature it is again 
necessary to refer to the properties of some arbitrarily-chosen substance 
or substances. In the construction of a scale of temperature it is cus- 
tomary to fix two points by means of well-defined and presumably 
constant properties of some standard substance, and divide the range 
between these two points into equal arbitrary units as measured by the 
change in property of some substance caused by change of temperature. 
Thus in the centigrade scale the two points which are fixed are the 
freezing point of the standard substance water, and secondly, its boiling 
point under a pressure of 76 cm. of mercury. When we use a mercury 
thermometer it is assumed that equal changes in volume of the mercury 
correspond to equal changes of temperature, 1 so by dividing the whole 
change of volume of the mercury between the freezing and boiling 
points of water into 100 equal portions, we obtain an ordinary centi- 
grade thermometer. Similarly, if we use an air thermometer, we 
assume that equal increments of volume correspond to equal increments 
of temperature. In each case we depend for our measurement of tem- 
perature on the properties of some particular substance, and it does 
not at all follow that the temperature as measured by one substance 
will exactly coincide with the temperature as measured by another. 
In point of fact, the temperatures registered by the mercury and air 
thermometers never exactly agree except at the point at which the two 
thermometers were originally made to correspond when their fixed 
points were determined. In theoretical calculations absolute tem- 
peratures are frequently employed. The absolute scale of tempera- 
ture has degrees of the same size as centigrade degrees, but starts from 
a point 273 degrees below zero centigrade. The absolute temperature 

1 In mercury and other liquid thermometers we really deal with the difference of the 
volume change of the liquid and of the vessel containing it. 


is therefore obtained by adding 273 to the temperature in the centi- 
grade scale. 

The usual unit of heat for chemical purposes is the "small calorie >J 
or gram calorie. It is roughly defined as the quantity of heat required 
to raise the temperature of one gram of water through one degree centi- 
grade. This quantity is not exactly the same at all temperatures, e.g. 
the amount of heat necessary to warm a gram of water from 0° to 1° 
is somewhat different from that required to heat it from 99° to 100°. 
It is therefore necessary to specify the temperature at which the heat- 
ing takes place. Practically the calorie measured at the ordinary 
temperature, say from 15° to 16°, is the most convenient. The great or 
kilogram calorie is 1 000 times the small calorie. A centuple calorie 
has also been proposed, and its definition is a practical one. It is the 
quantity of heat required to raise a gram of water from 0° to 100°, and 
is very nearly equal to 100 small calories. It is usually denoted by 
the symbol K, and we have thus the following relation of heat units : — 
1 Cal. = 10 K=1000cal. 

It is a matter of importance to fix the relation of the mechanical 
unit of energy to the heat unit, i.e. to determine the mechanical 
equivalent of heat. Mechanical energy, e.g. the energy of a falling 
body, is converted by friction or otherwise into heat, the amount of 
mechanical energy and the amount of heat resulting from it being both 
measured. In this way it has been found that 42,350 gram centi- 
metres are equivalent to 1 gram calorie, i.e. 42,350 grams falling 
through a centimetre will generate enough heat in a friction apparatus 
to raise the temperature of a gram of water from 0° to 1°. In the 
sequel we shall denote this value by J". 

The relations between the units employed in electrical measure- 
ments have been the subject of many accurate experimental investiga- 
tions. The units have been chosen theoretically so as to give a unit 
of electrical energy which bears a simple relation to the absolute unit 
of mechanical energy. The theoretical unit of resistance, the ohm, 
may be practically defined as the resistance of a column of mercury 1 
sq. mm. in section, and 1 *0626 metres long. A convenient practical unit, 
the Siemens mercury unit, is the resistance of a similar column exactly 
a metre long. The unit quantity of electricity, the coulomb, can be 
defined as the quantity of electricity which will deposit 1*118 mg. of 
silver from a solution of a silver salt under properly chosen conditions. 
The unit of potential difference or electromotive force, the volt, is so 
related to the previous units that when the potential difference between 
the ends of a resistance of 1 ohm is 1 volt, a current of 1 coulomb per 
second (1 ampfere) will pass through the resistance. For purposes 
of comparison the difference of potential between the electrodes of a 
normal galvanic cell is used as standard. Thus a Clark's cell, which 
is frequently employed for this purpose, has a potential difference 
between its electrodes equal to 1*434 volts at 15°. The unit of 


electrical energy, the volt-COTllomb, is the product of the volt into 
the coulomb, and is equivalent to 10 7 ergs, 10,204 gravitation units, 
or 0*241 cal. 

In connection with the subject of units, the student is strongly recom- 
mended to read Clerk-Maxwell, Theory of Heat, Chaps. I.-IV. 



Chemists have for long been in the habit of expressing their 
experimental results in terms of the Atomic Theory propounded 
by Dalton at the beginning of this century. At the time of its 
inception the facts which this theory had to explain were comparatively 
few in number and simple in character. Nowadays, the experimental 
data are numberless, and often of great complexity, yet the atomic 
theory is still capable of affording an easy and convenient mode of 
formulation for them, and is therefore to be pronounced a good 
theory. It is true that in the course of years the notions associated 
with the term " atom " have undergone many changes, but Dalton's 
fundamental idea is unaltered, and seems likely to persist for many 
years to come. 

The results of chemical analysis show that most substances can 
be split up into something simpler than themselves, but that a few, 
some seventy, resist all attempts at decomposition, and are incapable 
of being built up by the union of any other substances. These 
undecomposed bodies we agree to call elements, and to express the 
composition of all other bodies in terms of them. It does not, of 
course, follow that because we have hitherto been unable to decompose 
these elements methods may not at some later time be found for 
effecting their decomposition. In the sequel we shall see, however, 
that the elements form a group of substances, singular not only with 
respect to the resistance which they offer to decomposition, but also 
with respect to certain regularities displayed by them and not shared 
by the substances which go under the name of compounds. 

The atomic theory as at present understood affords a simple 
explanation of the manner in which elements unite to form compounds. 
Each element is assumed to be not infinitely divisible, but to consist 
rather of minute indivisible particles — the atoms — all alike amongst 
themselves, especially with respect to their weight. The atoms of 
different elements are supposed to differ in weight and in other 
properties. Compounds must, on our assumption, be built up of 


these atoms, the mode of their hypothetical union being chosen in 
accordance with the observed facts. In the first place, the atoms 
must be supposed to retain their weight unchanged, no matter in 
what state they may exist ; for all our chemical experience shows that 
chemical change is unaccompanied by any change in the total weight 
(or the total mass) of the reacting substances. Again, the same kinds 
of atoms in the same proportions go to the formation of any particular 
compound. This must be assumed to bring the theory into harmony 
with the constant composition exhibited by all compounds. Further, 
as the number of compounds actually existing is vanishingly small 
compared with the possible combinations of the different atoms, various 
rules have had to be devised regarding the nature and relative number 
of atoms which may unite with each other, in order that under these 
restrictions the number of possible combinations may agree more 
closely with the number of existing compounds. In these rules of 
"valency," etc., other properties of atoms besides their weight are 
indicated, and of late years it has been found necessary with regard 
to the carbon atom in particular to make very specific assumptions 
as to its nature, and the nature of its mode of combination with 
other atoms. 

From what has just been said, it is evident that our conception 
of the atom is made to accommodate itself to the facts it has to explain. 
Beyond the constancy of weight of each atom, nothing had to be 
assumed in Dalton's time except that the atoms united in proportions 
expressible by the simple whole numbers, and these assumptions were 
made in order that the theory might be in harmony with the laws 
of constant and of multiple proportions. It should be noted that at 
the present time we can scarcely speak of a law of simple multiple 
proportions existing. We know well - defined substances whose 
composition we are obliged to express by means of such formulae 
as C 20 H 23 N, C 13 H 14 5 — to choose two examples at random from 
organic chemistry. Here there is no approximation to simple 

When Dalton's theory had led to the adoption of a simple system 
for expressing the numerical proportions by weight in which elements 
combine and substances in general enter into chemical action, the 
conception of atom was practically allowed to drop, and its place 
was taken by the purely numerical conception of equivalent weight 
or combining proportion. The necessity of attaching any other 
significance to the atom than a constant weight was not generally 
felt, as the facts were not sufficiently well known to require any 
extension of the idea. Although the name " atom " continued to be 
used, the conception was practically that of a number. Laurent, 
for instance, in his Chemical Method (1853), says: "By the term 
* atoms,' I understand the equivalents of Gerhard t, or, what comes" to 
the same thing, the atoms of Berzelius." Again, " In order to avoid all 


hypotheses, I shall not attach to the term l atom ' any other sense 
than that which is included by the term l proportional number/ " 

For each element there existed a number which expressed its 
combining weight, and the composition of any compound was ex- 
pressed in terms of these weights as standards. It was therefore 
of extreme importance to fix the standard weights relatively to each 
other, in order that no dubiety or ambiguity should exist in the 
formulation of the compounds. Had there been no law of multiple 
proportions the task would have been easy. The weight of one 
element would have been chosen as ultimate standard to which all 
others would be referred, and as in that case each element would have 
united with each other element in one proportion only, no ambiguity 
could possibly exist. But the law of multiple proportions expresses 
the fact that a given^w^ight of one element may unite with more 
than one weight of another element to form more than one compound, 
and that then the ratios of the weights of the second element are 
expressible by means of the simple whole numbers. Taking the 
combining weight of hydrogen as standard and unit, the combining 
weight of oxygen might be either 8 or 16, for 1 g. of hydrogen unites 
with 8 g. of oxygen to form water, and with 1 6 g. to form hydrogen 
peroxide. So it is with most of the other elements. The choice 
becomes still more complex when the element to be considered forms 
no compound with the standard element hydrogen (as is often the 
case) and various compounds with the element oxygen, itself of 
doubtful combining weight. Some principle had therefore to be 
adopted for obtaining a consistent system of combining weights, only 
one for each element, to which all other weights might be referred. 

The extraordinary insight and acumen of Berzelius enabled him 
to arrive at a system of numbers not greatly differing from those 
in use at the present day, although he had little to guide him but 
the principles of analogy and simplicity of formulation. A better 
guide was gradually recognised in the uniformity displayed by sub- 
stances in the gaseous state. 

When the atomic theory was first propounded, no distinction was 
made between the ultimate particles of elements and the ultimate 
particles of compounds ; both were alike called atoms. By the atom 
of a compound was meant the smallest particle of it that could exist, 
any further subdivision resulting in the splitting of the compound into 
its elements, or into simpler compounds. It referred, therefore, to an 
ultimate particle which might be decomposed chemically, but not 
mechanically. On the other hand, the ultimate particle of an element 
was an atom which could not be decomposed either mechanically or 
chemically. There was thus a slight difference in the sense in which 
the term " atom " was used in regard to the two classes of substances ; 
and until this difference was recognised, little help was afforded to 
systematic chemistry from the study of gases. It had been observed 


by Gay-Lussac that gases entered into chemical actions in simple 
proportions by volume, the volumes of the different gases being 
of course measured under the same external conditions of tempera- 
ture and pressure. Now, according to Dalton, gases (like all 
other substances) also entered into chemical action in simple pro- 
portions by atoms. There was thus necessarily a simple connection 
between the atoms of gases and the volumes they occupied. Dalton 
himself, led by other speculations, tried the assumption that equal 
volumes of different gases contained the same number of atoms, i.e. that 
the weights of equal volumes of different gases were proportional to 
the weights of their atoms. He rejected this supposition, however, as 
not being in harmony with the facts. Two measures of nitric oxide 
give on decomposition one measure of nitrogen and one measure of 
oxygen, i.e. on the above assumption, two <gtams of nitric oxide give 
one atom of nitrogen and one atom of oxygen. But there must be at 
least two atoms of nitrogen in two atoms of nitric oxide if the 
elementary atoms are really indivisible, which contradicts the above 
hypothesis. Avogadro showed how this difficulty might be overcome 
by distinguishing in the case of the elements between the different 
senses of indivisibility above referred to. The gist of his reasoning is 
as follows : — 

The particles of a gas may be supposed to be the smallest particles 
obtainable by mechanical division, whether they are particles of 
elements or of compounds. But it does not follow, even for the 
elements, that this limit of mechanical or physical divisibility is also 
the limit of chemical divisibility. The gaseous particles of a compound 
can be chemically decomposed into something simpler ; so may the 
gaseous particles of an element, only in the latter case the products of the 
decomposition of the gaseous particle must be atoms of the same kind, 
and not atoms of different kinds as with the compounds. He made 
the distinction, therefore, between atoms and molecules as these terms 
are used at the present day. The molecules are the mechanically 
indivisible gaseous particles, which may consist each of more than one 
elementary atom — of different kinds in the case of compounds, of the 
same kind in the case of elements. 

The hypothesis which Avogadro now made to account for the 
relations between combining weights and volumes was the following. 
Equal volumes of different gases under the same physical conditions 
contain the same number of molecules ; or, the weights of equal 
volumes of different gases are proportional to the weights of their 
molecules. There is now no longer any difficulty encountered in the 
volume relations of nitric oxide and its decomposition products. 
According to Avogadro, two molecules of nitric oxide give one 
molecule of nitrogen and one molecule of oxygen. But as two 
molecules of nitric oxide must contain at least two atoms of nitrogen 
and two atoms of oxygen, the molecule of nitrogen must contain at 


least two atoms of nitrogen, and the molecule of oxygen two atoms of 

The general adoption of this one principle has practically proved 
sufficient to fix the combining weights of the elements, and provide us 
with our present system of atomic weights. Although Avogadro 
published his hypothesis in 1811, the times were not ripe for it; and 
it was only in the fifties that its application became at all general. In 
the middle of the century the greatest confusion prevailed, different 
writers using different systems of combining weights, so that each 
compound had as many different formulae as there were competing 
tables of equivalents. It is only within a few years that the last 
traces of one of the old systems have disappeared, many chemists, 
especially in France, having adhered to equivalents which gave HO or 
H 2 2 as the formula of water, and C 2 H 4 as the formula of marsh gas. 
So long as weights only were considered, one system was as good as 
another; but when volume relations, physical properties, and the 
nature of the substances produced in chemical actions systematically 
carried out, are taken into account, the present system of atomic 
weights is the only one which gives a simple and consistent expression 
of the results. 

The hypothesis of Avogadro often goes under the name of 
Avogadro's Law, but it must be borne in mind that it is a purely 
hypothetical statement, and not to be confounded with a generalised 
expression of fact such as the Law of Constant Proportions. This 
latter is independent of any theory, while the former is entirely 
theoretical — a fact which of course in no way impairs its usefulness. t 

In the following pages a sketch is given of the modes employed in 
fixing our present system of atomic weights with the help of 
Avogadro's principle. The atoms and molecules having themselves 
only a hypothetical existence, we are not for our present purpose con- 
cerned with their absolute weight — all that we require is to express 
their weights relatively to each other. To begin with, a standard 
must be chosen to which all chemical weights are to be referred. The 
standard is purely arbitrary, and in choosing it we only consult our 
own convenience. Two elements, hydrogen and oxygen, have been 
selected on practical grounds for this purpose. Dalton chose hydrogen 
because it had the smallest equivalent of the elements ; Berzelius 
chose oxygen because it was easy to compare the equivalents of the 
other elements with that of oxygen directly. Hydrogen does not 
form many well-defined and easily-analysed compounds with the other 
elements, so that comparison with the equivalent of hydrogen has 
usually to be indirect, frequently through the medium of oxygen. 
Both of these standards are at present in use. Some chemists choose 
hydrogen, and make its atomic weight equal to 1. Others select 
oxygen, and fix its atomic weight at 16. The actual number selected 
as standard magnitude is again arbitrary ; Berzelius, for example, made 


the equivalent of oxygen equal to 100, and referred the other numbers 
to this. The reason why the number 16 has been chosen to express 
the atomic weight of oxygen when that element is taken as standard, 
is that when hydrogen is made to have an atomic weight equal to 
unity the number which then expresses the atomic weight of oxygen 
is very nearly equal to 16. Thus, whichever of the two standards is 
adopted, we obtain practically the same set of numbers for the atomic 
weights of the other elements, if round numbers only are required. 
For purposes requiring a high degree of accuracy it is necessary to 
distinguish between the two sets. Formerly the ratio between the 
atomic weights of oxygen and hydrogen was not known with the 
same degree of accuracy as had been attained in fixing the ratios of 
the atomic weight of oxygen to that of many other elements. By 
taking H = 1 therefore, each new determination of the ratio H : 
necessitated the alteration of a great many atomic weights. By taking 
= 16, the new determination only necessitates the alteration of the 
atomic weight of hydrogen. From recent researches carried out by 
independent investigators the ratio H : has been determined with 
accuracy to be 1 : 15*88, or 1*0075 : 16. With = 16 then, we have 
a set of atomic weights all 1*0075 times greater than when H=l. 
For accurate purposes this difference between the two systems must 
be considered, but for the everyday purposes of chemistry the system 
is practically one of round numbers based on 0=16, although we look 
on H = 1 as the theoretical standard. 

In fixing our system of atomic weights on Avogadro's principle, the 
weights of equal volumes of gases are compared. 1 Now the 
weights of equal volumes of gases are proportional to their relative 
densities, and for the present purpose these densities are most con- 
veniently referred to that of hydrogen = 2. The reason for choosing 
the density of hydrogen equal to 2 instead of equal to 1, as is custom- 
ary, is that on the former assumption the density of a gas is expressed 
by the same number as its molecular weight, instead of by half that 
number. The selection of the unit is quite arbitrary, and we may 
as well choose that which leads to the greatest simplicity. The atomic 
weight of hydrogen is 1, and on Avogadro's principle we must argue 
that the molecule of hydrogen contains two atoms, in order to satisfy 
the volume relations for the formation of hydrochloric acid gas from 
hydrogen and chlorine. Two volumes of hydrochloric acid gas are formed 
from one volume of hydrogen and one volume of chlorine, so that by 
reasoning similar to that employed above in the case of nitric oxide, 
each hydrogen molecule and each chlorine molecule must contain two 
atoms. The molecular weight of hydrogen is therefore at least 2 
if its atomic weight is 1. It is convenient thus to express molecular 
weights in the same unit as atomic weights, for then the molecular 

1 The volumes must of course be measured under the same conditions of temperature 
and pressure. 


weight is simply the sum of the atomic weights contained in the 

The determination of the molecular weight of a substance therefore 
resolves itself into finding the weight of its vapour in grams which will 
occupy the same volume as 2 g. of hydrogen measured under the same 
conditions of temperature and pressure. This weight is called the gram- 
molecular weight of the substance, and the volume which it occupies is 
called the gram-molecular volume. It is practically the same for all 
gases, and at 0° and 76 cm. may be taken equal to 22*38 litres. Only 
rough values of the molecular weight are in this way obtained, for the 
determination of the density of gases and vapours does not under 
ordinary circumstances admit of any great degree of accuracy ; besides 
which Avogadro's principle can be applied in strictness only to perfect 
gases, i.e. to those which obey the laws for gases with absolute exact- 
ness. It is easy, however, to arrive at the true molecular weight from 
the approximate value obtained from vapour-density determinations by 
taking account of the results of analysis, which are susceptible of great 
accuracy. It is evident that the true molecular weight must contain 
quantities of the elements which are exact multiples or submultiples of 
the combining proportions of these elements, and the combining pro- 
portions are the results of analysis alone. We therefore select as true 
molecular weight the number fulfilling this condition which is nearest 
the approximate molecular weight. For example, the molecular weight of 
sulphuretted hydrogen, as determined from the vapour density, is 34*4, 
i.e. 34*4 g. of sulphuretted hydrogen occupy the same volume as 2 g. of 
hydrogen under the same external conditions. But we know that for 
1 g. of hydrogen in the sulphide there are 16*0 of sulphur. The true 
molecular weight therefore must contain a multiple of TO g. of 
hydrogen, and the same multiple of 16*0 of sulphur. The number 
34*0 evidently does this, for it is equal to 2(1*0 + 16*0). We conse- 
quently take 34*0 as the true molecular weight of hydrogen sulphide 
instead of the number 34*4 obtained from the determination of the 
vapour density. 

The atomic weight of an element is now deduced from the true 
molecular weights of its gaseous compounds as follows. A list of the 
molecular weights of the gaseous compounds is prepared, and the 
quantities of the element in these weights of the compounds are noted 
alongside. The figures in this second column are the results of 
analysis alone, and their greatest common measure is in general equal to 
the atomic weight of the element. Examples are given in the follow- 
ing tables. The first column of numbers contains the gram-molecular 
weights of the compounds ; the second column contains the number of 
grams of the element in question in the gram-molecular weight ; the 
third column shows the relation of these numbers to their greatest 
common measure. 



Hydrochloric acid . 
Hydrobromic acid . 
Hydriodic acid 
Water . 

Hydrogen sulphide 

Hydrogen phosphide 
Ethane . 

Water . 

Carbon monoxide . 
Phosphorus oxychloride 
Nitric oxide . 
Carbon dioxide 
Sulphur dioxide 
Chlorine peroxide . 
Sulphur trioxide 
Methyl nitrate 
Osmium tetroxide . 

Hydrochloric acid . 
Chlorine peroxide . 
Nitrosyl chloride . 
Cyanogen chloride . 

Chlorine monoxide 
Thionyl chloride . 
Sulphuryl chloride . 
Phosphorus trichloride 
Phosphorus oxychloride 
Boron trichloride . 
Carbon tetrachloride 

Nitric oxide/ . 
Nitrogen peroxide 
Methyl nitrate 
Cyanogen chloride 
Nitrous oxide 


















G.C.M. I 




. 16 




































































G.C.M. 14 




Compound. I. II. III. 

Methane 16 12 1x12 

Chloroform 119*5 12 1x12 

Carbon monoxide 28 12 1 x 12 

Carbon dioxide 44 12 1 x 12 

Cyanogen chloride 61*5 12 1x12 

Ethylene 28 24 2x12 

Ethane 30 24 2 x 12 

Cyanogen 52 24 2x12 

Acetylene 26 24 2x12 

Propane 44 36 3 x 12 

Butane 58 48 4x12 

Pentane 72 60 5x12 

Hexane 86 72 6x12 

Benzene 78 72 6x12 




Hydrogen sulphide 




Sulphur dioxide 




Sulphur trioxide . 




Sulphuryl chloride 

. 135 







Carbon disulphide . 




G.C.M. 32 

Proceeding on the principle explained above, we should now make 
the following table of the atomic weights of these elements : — 

Hydrogen = 1 
Chlorine =35*5 
Nitrogen =14 
Carbon =12 
Sulphur =32 

These numbers are the generally accepted values for the atomic 
weights of the elements above considered, and indeed if any element 
has a large number of volatile compounds whose molecular weights 
can be determined from their vapour densities, the principle we have 
adopted leads to practically certain conclusions. The weights of any 
element contained in the molecular weights of its compounds must be 
equal to the atomic weight or must be multiples of it, so that if we 
take the greatest common factor of these multiples, it must either be 
a simple multiple of the atomic weight or the atomic weight itself. 
From this it appears, then, that the method is one which only gives values 
which the atomic weights cannot exceed. But if a great many volatile 
compounds of the element are known, the chance is very slight 
that the greatest common factor is still a multiple of the atomic 
weight, i.e. that another substance may be discovered which will not 
contain this greatest common factor in the weight of the element 
which enters into its molecule ; and so the atomic weight fixed in the 


manner indicated above has a great degree of probability. If, on 
the other hand, only a few volatile compounds of the element are 
available for vapour- density determinations, the method may con- 
ceivably fail to give the correct result. Nowadays, however, we are 
in possession of other means of determining molecular weights besides 
the method of vapour densities, so that for each element a great 
many compounds of known molecular weight can be tabulated, even 
though the volatile compounds of some may be few in number. If 
we make use of these new methods, the atomic weights of the elements 
as determined from the molecular weights of their compounds are 
practically fixed with certainty. 

The recently -discovered gases, argon, helium, etc., are in all 
probability elementary ; but it has not been found possible to 
determine their atomic weights with any certitude of finality. They 
enter into no combinations, so that no analysis is possible, and the 
only method for ascertaining their characteristic weights is the deter- 
mination of their vapour densities. Their atomic weights must in 
the meantime be assumed equal to their densities compared with 
hydrogen equal to 2, i.e. their molecules must be assumed to consist 
of one atom until we find reason to the contrary. That this as- 
sumption has a certain degree of probability will be seen in the 
sequel (Chap. V.) 

Other methods have been used for fixing the atomic weights of 
the elements, but these must now be looked on rather as checks on 
the method from molecular weights than as methods of independent 
applicability. These are the methods depending on Dulong and 
Petit's Law (Chap. V.), and on the Periodic Law (Chap. VI. ), and 
will be referred to in the sections treating of these regularities. 

From what has been said, the fixing of exact values of the 
atomic weights consists of two problems — the determination of a set 
of equivalents from accurate quantitative experiments, and then the 
selection of one of these according to some definite principle such 
as that referred to in the preceding pages. The nature of the 
quantitative experiments performed depends on the character of the 
element whose equivalent is to be determined, but it always consists 
in the exact comparison of the weights of two substances which contain 
the same quantity of the element. As far as possible, the experiments 
are chosen of a simple kind, and should involve the smallest possible 
number of other elements. Berzelius, for example, dissolved 25*000 
g. of lead in nitric acid, evaporated the lead nitrate to dryness 
and ignited the residue carefully, thereby obtaining 26*925 g. of 
lead oxide. Twenty-five g. of lead combine therefore with 1*925 g. 
| of oxygen, so that if we take = 16, we obtain the value 207*8 as 
.one of the possible equivalents of lead. Taking the mean of four 
(similar experiments, we obtain the value 207*3. These experiments 
may, however, all be affected by a systematic error, i.e. by an error 


which is the result of this particular method of experiment. The 
equivalent must therefore be determined by some other method as a 
check. Berzelius converted lead directly into lead sulphide, and 
obtained 207*0 as the combining weight of lead if that of sulphur was 
taken as 32. By converting weighed quantities of lead oxide into 
lead sulphate, he obtained Pb = 207*0, on the assumption 0=16 and 
S = 32. Other observers using different methods have all got 
numbers approximating to 207*0, and Stas, who used every conceivable 
precaution, and whose number is usually accepted as the most correct 
value, obtained 206*9. 

The error in a well-conducted quantitative transformation under 
ordinary laboratory conditions is about 0*1 per cent of the amount 
transformed. To secure an error as low as 0*01 per cent extraordinary 
precautions have to be taken, and it is only in the hands of skilled 
operators that this degree of accuracy can be attained. In order, too, 
that this accurate work may be of value, it is obviously essential that 
the substances operated on should be of a degree of purity corre- 
sponding to the excellence of the quantitative determinations. Now 
" chemically pure " substances usually contain an amount of impurity 
greatly exceeding 0*01 per cent of their weight, so that special skill 
and trouble must be devoted to the purification of the substances 
employed in the determination of the gravimetric ratio. Even after an 
experimental accuracy of 0*01 per cent has been secured with pure 
substances, the error in the equivalent is generally much larger than this. 
Thus, supposing that Berzelius made an error of only this amount 
in the conversion of lead into lead oxide, 25 g. of lead would give 
26*925 g. of oxide, with a possibility of error of 0*0025 g. This 
means, however, that the quantity of oxygen with which the lead 
has combined is 1*925, with a possible error of 0*0025 g., ie. an error 
of 0*12 per cent on the amount of oxygen. Now this amount is 
required to fix the equivalent of lead, so that the error in the 
equivalent is not less than 0*12 per cent, and the value 207*8 may 
therefore be two units out in the first decimal place, even on the 
assumption of this very great experimental accuracy. 

It may be taken for granted that only for very few elements has 
an accuracy of 0*1 per cent on the atomic weight been attained. The 
elements whose atomic weights were determined by Stas with every 
experimental refinement are generally accepted as being the most 
accurate. 1 As an example of modern work, we may take recent 
determinations of the ratio in which oxygen and hydrogen combine to 
iorm water, a very important ratio, since these substances are the 
standards of the different systems of atomic weights. Here gases have 
to be weighed, a fact which greatly enhances the experimental difficulty, 
so that the highest degree of accuracy can scarcely be expected. 

1 These are silver, the halogens, potassium, sodium, lead, sulphur, with less accurate 
determinations for lithium and nitrogen. 


Observers ' Atomic Weight of 

UDseners. ; Hydrogen for = 16. 

Dittmar and Henderson . . I 1 '0085 

Scott and others 1 '0082 

Cooke and Richards .... 1 -0082 

Morley ! 1*0075 

Leduc 1-0074 

Noyes 1*0064 

Keiser i 1*0031 

Atomic Weight of 
Oxygen for H = l. 


If we exclude Reiser's number, which diverges rather widely from 
the others but has not been proved directly to be inaccurate, the 
extreme difference between the values of different observers is 0*2 per 
cent — an exceedingly good agreement. The value generally accepted 
at present for the atomic weight of hydrogen is the mean 1*0075, or 
1*01 for accurate practical purposes, although in the laboratory the 
approximation 1*0 is usually sufficient. 

The following table of atomic weights has been adopted by a com- 
mission of the German Chemical Society (1898) as giving the most prob- 
able values for practical use, and may be taken to that extent as 
official. Each figure given is significant. Thus, the atomic weight 
of carbon, C= 12*00, indicates that the atomic w r eight probably lies 
between 11*995 and 12*005 ; whilst the atomic weight of boron, B = 
11, indicates that the true atomic weight probably lies between 10*5 
and 11*5. It is possible that the atomic weights of some elements 
have been determined with greater accuracy than that given, but the 
reverse is probably more often the case. 








Neodymium (?) 









Argon (?) 






































79 '96 





































Samarium (?) 





















Erbium (?) 






























Helium (?) 











































































* The figure after the decimal is very doubtful. 

An inspection of this table shows that there are some forty 
elements whose atomic weights have been determined with such 
accuracy that the first figure after the decimal point may be supposed 
to possess a real significance. Now according to the rules of prob- 
ability, we might expect one-fifth of these, namely eight, to have values 
of the first decimal figure lying within 0*1 of a whole number. In- 
stead of one-fifth, however, we find that one-half of this number of 
elements, namely twenty, possess atomic weights diverging by 0*1 or 
less from the whole numbers. If we only consider elements whose 
atomic weights are less than 100, the chances of accuracy of the first 
decimal place are increased, since a unit in that place forms a higher 
proportion of the whole ; yet here also the proportion of elements 
diverging by 0*1 at most from whole numbers is considerably more than 
half ; and if we consider only the element whose atomic weights were 
determined by Stas, we find that two-thirds of them have atomic weights 


diverging by less than 0*1 from whole numbers. It is difficult to believe 
that this approximation to whole numbers is quite fortuitous. Early in 
the century Prout suggested that all atomic weights were multiples of 
that of hydrogen. This is clearly not the case, for well-investigated 
elements like chlorine and copper have atomic weights which cannot 
possibly be brought under such a scheme. It is much less easy to 
prove that the atomic weights of the elements are not multiples of the 
thirty -second part of the atomic weight of oxygen, i.e. that all atomic 
weights do not end in '0 or *5. There is a distinct grouping of the 
values of the first decimal place round these two numbers, and the 
grouping is more marked the more we exclude elements of doubtful 
atomic weight. To conclude from this, however, as has sometimes been 
done, that all the elements are built up of a fundamental element whose 
" atomic weight " is one-half of that of hydrogen, and that the actual 
divergencies from whole or half numbers are due to experimental error, 
is, to say the least, scarcely justifiable in the present state of our know- 

An interesting account of the work of Stas, and a general discussion of 
methods of atomic-weight determination, will be found in the " Stas Memorial 
Lecture" by J. W. Mallet (Journal of the Chemical Society, vol. lxiii. p. 1). 
Some work by Stas on pure chemicals is reprinted in the Chemical News, vol. 
lxxii. The student may also be referred to the following examples of modern 
determinations of equivalents : — 

T. W. Richards (American Academy of Arts and Sciences, vols. xxv. to 
xxviii., and Chemical News, vols, lxiii. to lxvii.) : Equivalents of Copper and 
of Barium. 

E. W. Morley (Smithsonian Contributions, vol xxix., and Zeitschrift fur 
physikalische Chemie, vol. xx.) : Combining Ratio of Oxygen and Hydrogen. 



In general, if we know all the reacting substances in a chemical 
transformation, and all the products of the reaction, there is one and 
only one numerical solution of which the equation is susceptible. For 
example, if we are given the reacting bodies, copper and nitric acid, 
and the products, copper nitrate, water, and nitric oxide, i.e. the in- 
complete equation 

? Cu + 1 HN0 3 = 1 Cu(N0 8 ) a + ? H 2 + ? NO, 

there is only one set of numbers (not containing a common factor) by 
means of which we can complete the equation, viz. 

3Cu + 8HNO3 = 3Cu(N0 3 ) 2 + 4H 2 + 2NO. 

It is occasionally a matter of some difficulty to find the proper numbers, 
and the student is not recommended to proceed to the solution by a 
process of trial and error, for even in a case like the above the work 
entailed would be considerable. The solution can always be arrived 
at by splitting the reaction ideally into simpler reactions whose 
numbers can be easily determined by inspection, and adding together 
the equations thus obtained. With a little practice the student can 
soon acquire sufficient skill to enable him to dispense with the 
cumbrous array of figures which he must otherwise carry in his 

Taking the above example, we might conceivably attack the 
problem in two different ways. The essential feature of the action 
is reduction of the nitric acid and oxidation of the copper. We begin, 
then, by writing an equation to express the reduction of the nitric acid 
to nitric oxide, ascertaining how many atoms of oxygen are available 
for oxidising purposes : — 

2HN0 3 = H 2 + 2NO + 30. 
It is evident that the numbers must be as they are given here in order 


that we may have the same numbers of atoms of hydrogen and nitrogen 
on the right-hand side. 

Then the oxidation of the copper by the three atoms of oxygen is 
expressed thus : — 

3Cu + 30 = 3CuO, 

and the conversion of the copper oxide into nitrate by the nitric acid 
thus : — 

3CuO + 6HNO3 = 3Cu(N0 3 ) 2 + 3H 2 0. 

Adding these equations together, right and left, we get 

3Cu + 8HNO3 = 3Cu(N0 3 ) 2 + 2NO + 4H 2 0. 

We might, instead of considering the copper to be oxidised directly to 
the oxide, imagine it to be converted into nitrate by the nitric acid, 
with liberation of hydrogen, the hydrogen then going to reduce nitric 
acid to nitric oxide, or we might write everything in the old dualistic 
system, which supposes salts to be made up of an acid and a basic oxide, 
and acids to be composed of an acid oxide and water, thus : — 

2HN0 3 = N 2 5 + H 9 0, 
N 2 5 = 2NO + 30, 
3Cu + 30 = 3CuO, 
3CuO + 3(N 2 5 . H 2 0) = 3(CuO . N 2 5 ) + 3H 2 0. 

Adding these equations, and rejecting the terms which appear on both 
sides, we get the same equation as before. 

As another instance, we might take the oxidation of ferrous 
chloride to ferric chloride by potassium bichromate in presence of 
hydrochloric acid : — 

1 K 2 Cr 2 7 + ? FeCl 2 + ¥ HC1 = 1 KC1 + ? CrCl 3 + ? FeCl 3 + ? H 2 0. 

One side of the action is 

FeCl 2 + HCI = FeCl 3 + H. 

The bichromate is converted into potassium chloride, chromic chloride, 
and water : — 

K 2 Cr 2 7 + 8HC1 = 2KC1 + 2CrCl 3 + 4H 2 + 30. 

The surplus oxygen unites with the hydrogen of the first action, which 
necessitates multiplying its equation by 6, thus : — 

6FeCl 2 + 6HC1 = 6FeCl 3 + 6H, 
6H + 30 = 3H 2 0. 

Adding these three equations, we obtain 

K 2 Cr 2 O r + 6FeCl 2 + 14HC1 = 2KC1 + 2CrCl 3 + 6FeCl 3 + 7H 2 0. 


Sometimes equations appear indeterminate, and permit an indefinite 
number of numerical solutions. An example may be found in the 
decomposition of potassium chlorate by heat, with formation of 
potassium perchlorate, potassium chloride, and oxygen : — 

? KCIO3 = ? KC10 4 + I KC1 + ? 2 . 
The equation usually given in the text-books is 

2KC10 3 = KC10 4 + KCl + 2 , 
and this has the merit of simplicity. But the equation 

6KCIO3 = KC10 4 + 5KC1 + 70 2 

is numerically quite as correct as the other, although entirely different 
from it, and there are besides an infinite number of other solutions. 
The reason for this divergence from the general rule that there is only 
one solution to each equation, is that we are here dealing with two 
apparently independent actions which go on simultaneously, viz. — 

(a)2KC10 8 = 2KCl + 30 gJ 

(6)4KC10 3 = KC1 + 3KC10 4 . 

If we add these equations as they stand, we get 

or dividing by 3, 

6KCIO3 = 3KC1 + 3KC10 4 + 30 2 , 
2KC10 S = KC1 + KC10 4 + 2J 

the usual book equation. 

Multiplying the equation (a) by 7, thus — 

14KC10 3 =14KC1 + 210 2 , 

and adding (b), we obtain 

18KC10 3 = 15KC1 + 3KC10 4 + 210 2 . 

Dividing by 3, we get 

6KC10 3 = 5KC1 + KC10 4 -f 70 2 , as above. 

In general, multiplying (a) by x, (b) by y, and adding, we have the 

2(x + 2#)KC10 3 = (2x + y)KCl + Sx0 2 + 3yKC10 4 , 

where x and y may each independently have any integral value 
whatever, so that the numerical solutions are infinite in number. 
That we are here actually dealing with two simultaneous reactions is 
evident from the fact that the relative proportions of chloride, per- 
chlorate, and oxygen formed depend on the temperature and the 
conditions employed to effect the decomposition. . 

Another instance may be found in the action of hydrochloric acid 
on a chlorate to produce " euchlorine " : — 

% KC10 3 + ? HC1 = ? KC1 + % H 2 + 1 C10 2 + % Cl 2 . 


This equation too admits of an indefinite number of solutions. In 
reality two reactions progress simultaneously, viz. — 

(a) KCIO3 + 6HC1 = KC1 + 3H 2 + 3C1 2 , 
(b) 5KC10 3 + 6HC1 = 5KC1 + 3H 2 + 6C10 2 . 

If we add these equations, we obtain 

6KC10 3 + 12HC1 = 6KC1 + 6H + 3C1, + 6C10 2 , 
or 2KCIO3 + 4HC1 = 2KC1 + 2H 2 + Cl 2 + 2C10 2 . 

The equation usually given in the text-books is 

8KCIO3 + 24HC1 = 8KC1 + 12H 2 + 6C10 2 + 9C1 2 , 

which is more complicated. It may be obtained by multiplying (a) 
by 3 and adding it to (b), thus : — 

3KC10 3 + 18HC1 = 3KC1 + 9H 2 + 9C1 2 

5KC1Q 3 + 6HC1 = 5KC1 + 3H 2 Q + 6C1Q 2 

8KCIO3 + 24HC1 = 8KC1 + 1 2H 2 + 6C10 2 + 9C1 2 . 

The general expression is 

(x + 5y)KC10 3 + 6(s + y)HCl = (x + oy)KCl + 3 (a; + y)R 2 + 3:rCl 2 

+ 6yC\Q 2 , 

where x and y may be given any integral values whatever. The 
actual proportion of chlorine and chlorine peroxide obtained in any 
one experiment depends on the temperature and on the concentration 
of the hydrochloric acid employed. From a consideration of the 
general equation it is easy to obtain an equation which shall express 
the relative proportions of Cl 2 and C10 2 obtained in a given experiment. 
Thus, to write an equation which will correspond to the case of 
equal volumes of chlorine and chlorine peroxide being evolved, we 
have only to make x= 2 and y=l, which will give the same number 
of molecules of chlorine and chlorine peroxide, and therefore equal 
volumes, thus : — 

7KC10 3 + 18HC1 = 7KC1 + 9H 2 + 6C1 2 + 6C10 2 . 

The reduction of nitric acid to form more than one of the lower 
oxides of nitrogen will afford similar instances to the student, who is 
advised to make himself familiar with them. 

To express the formation of phosphonium iodide and hydriodic 
acid from phosphorus, iodine, and water, we sometimes find in books 
the following complicated equation : — 

91 + 13P + 24H 2 = 6H 3 P0 4 + 2HI + 7PH 4 L 

That this is needlessly complex may be seen by simple inspection, for 
without any difficulty we can arrive at the solution 

21 + 2P + 4H 2 = H 3 P0 4 + HI + PH 4 L 


Since the equation admits of two numerical solutions it must be 
indeterminate, and permit of an infinite number. We can get the 
general solution as usual by splitting up the indeterminate equation 
into the two independent determinate equations of which it is com- 
posed. In the one case we have the production of phosphoric acid 
and hydriodic acid from the reacting substances ; in the other we 
have the production of phosphoric acid and phosphonium iodide, the 
determinate equations for these reactions being 

51 + P + 4H 9 = H 3 P0 4 + 5HI, 
51 + 9P + 16H 2 = 4H 3 P0 4 + 5PH 4 I. 

The general equation is then 

5(x + y)l + (x+ 9?/)P + i(x + 4#)H 2 = (x + 4^)H 3 P0 4 + 5zHI + 5yPH J. 

By making x = 1 and y = 1 in this equation, we obtain, on simplifying, 
the solution 

21 + 2P + 4H 2 = H 3 P0 4 + HI + PH 4 I, 

and by making x = 2 and y = 7, we obtain the solution 

91 + 13P + 24H 2 = 6H 3 P0 4 + 2HI + 7PH 4 L 

Occasionally it happens that substances having the same composi- 
tion, i.e. isomeric or polymeric substances, are produced by a chemical 
action. In this case the equations are indeterminate, although the 
chemical action itself may be quite definite. Thus, for the inversion 
of cane sugar we have the equation 

1 C 12 H 22 O u + 1 H 2 - 1 C 6 H 12 6 + t C 6 H 12 6 . 

(dextrose) (levulose) 

This equation is evidently susceptible of an indefinite number of 
solutions, if we consider separately the isomeric molecules produced, 
although the only chemical action that goes on is expressed by the 
equation giving the same number of molecules of dextrose and 
levulose, viz. 

^12 H 220]l + H 2^ = ^6^12^6 + ^0^12^6' 

(dextrose) (levulose) 



The student is doubtless familiar with the fact that the laws regulat- 
ing the physical condition of gases are of an extremely simple character 
and of universal applicability. Pressure and temperature affect the 
volume of all gases to nearly the same degree, no matter what the 
chemical or other physical properties of the gases may be, so that we 
can state for gases the following general laws : — 

1. Boyle's Law. — The volume of a given mass of any gas varies in- 
versely as the pressure upon it if the temperature of the gas remains 

2. Gay-Lussac's Law, — The volume of a given mass of any gas 
varies directly as the absolute temperature of the gas if the pressure 
upon it remains constant. 

3. The pressure of a given mass of any gas varies directly as the 
absolute temperature if the volume of the gas remains constant. 

Any one of these laws may be deduced from the other two, and 
as an example we may take the deduction of the third law from the 
laws of Boyle and Gay-Lussac. Let the original pressure, volume, and 
temperature of the gas be p 0i v Q) and T . Let the gas be heated at the 
constant pressure p to the temperature T v Then, according to Gay- 
Lussac's Law, 


if V is the volume assumed at T v 

Now let the gas be compressed to the original volume v at the 
constant temperature T v According to Boyle's Law, the product of 
the pressure and volume of a gas always remains the same if the tem- 
perature is constant. We have therefore at T 1 

p F=p 1 v , 

if p 1 represents the value of the pressure after compression to the 
volume v . But from the first equation we have 



Substituting this value in the second equation, we obtain 

pi n 

which is an expression of the third law for the variation of pressure 
with temperature at constant volume. 

These three separate laws may be put into the form of a single 
equation as follows. Let the pressure, volume, and temperature of a 
mass of gas all change, the original values being p , v , T QJ and the 
final values p v v v T v Suppose first the volume to remain constant at 
v Qi and the temperature to assume its final value T v The gas will then 
have a pressure P, given by the equation 

^> = Zo 
P T{ 

Let the gas now expand from v to v v at constant temperature 2^; the 
new pressure p x will then be given by the equation 

Pv =pjV v 


F= Pfi. 

Substituting this value of P in the pressure-temperature equation, we 
have finally 

m T i 

i.e. the product of the pressure and volume of a gas is proportional to 
the absolute temperature. 

The final result may also be made to assume the form 

that is, the expression pv/T for a given quantity of gas always remains 
constant. Since if pressure and temperature are constant the quantity 
of a gas is proportional to the volume taken, the actual value of the 
expression pv/T is proportional to the quantity or volume considered. 
Now, since according to Avogadro's principle the gram-molecular 
weights of all gases occupy the same volume under the same condi- 
tions of temperature and pressure, it follows that for these quantities 
the expression will have a constant value, no matter what the nature 
of the gas is, or the conditions under which the volume is measured. 
We have therefore in general for all gases 


PV_ P 

where 11 is a constant, and V the volume occupied by the gram mole- 
cule of the gas. At 0°and 76 cm. the gram-molecular volume is 22*4 
litres, or, more exactly, 22,380 cc. (cp. p. 14). The pressure p 
in grams per square centimetre is 1033 (p. 3), and T is 273. To 
calculate the value of R we have therefore 

^ = 1033^380 = g4678 

This value of the general gas constant R bears a very simple relation 
to J, the mechanical equivalent of heat, and advantage may be taken of 
this circumstance to throw the gas equation into a very convenient 
form for many calculations. 

The product pv is of the dimensions of mechanical energy, and may 

be expressed in gram centimetres, p being expressed as -^— v and v as 

cm. 3 , the product therefore being g. x cm., or gram centimetres. To 
convert this product into thermal units we divide by /= 42,350, and 
thus obtain 

1033 x 22,380_ 84,678 _ 9 
*"" 273x42,350 "42,350""* 

The general gas equation for molecular quantities may therefore 
be written in the form 

when thermal units are employed. 

This equation is very frequently used in measuring the work 
done upon or done by a gas when it changes its volume under external 
pressure. The amount of work is equal to the product of the pressure 
into the change of volume, and can be easily expressed in thermal 
units by means of the above equation when molecular quantities are 
considered. Thus if a gram molecule of a gas is generated by chemical 
action, say from the interaction of zinc and an acid, an amount of heat 
will be absorbed in performing the external work of expansion equal 
to 2 T calories, i.e. to 546 calories if the action takes place at 0° C. 
Other examples of the use of this equation will be found in the 
chapters dealing with specific heats, the gas laws for solutions, and 
more especially in the chapter on thermod} 7 namics. 

These simple gas laws, and the deductions from them, apply in 
strictness only to ideal or " perfect " gases, no actually existing gas 
obeying them exactly. For most purposes of calculation, however, 
they may be applied to ordinary gases without any very serious error 
being committed, if care be taken in applying them to vapours near 
the point of condensation or to gases under very great pressures (cp. 
Chapters IX and X.) 



As early as 1819, Dulong and Petit noticed that the products of the 
atomic weights of the elements into their specific heats were approxi- 
mately constant. Of course at that time the atomic weights of the 
elements had been determined with very little certainty, but taking 
the atomic weights even as they stood there was an unmistakeable 
appearance of regularity. In cases where there was doubt as to which 
equivalent of an element ought to receive the preference, the rule given 
by Dulong and Petit was helpful. It was evidently more rational to 
choose that equivalent which conformed to the regularity displayed 
by the other elements than an equivalent which would make the 
element under discussion appear exceptional. This means of selecting 
a standard number from amongst the different possible equivalents of 
an element to represent its atomic weight proved of great use during 
the period when Avogadro's principle was not recognised as a safe 
guide in fixing molecular and atomic quantities ; and even after the 
recognition of this principle, the Law of Dulong and Petit rendered 
much service in precisely those cases where the application of 
Avogadro's Law failed from want of data. 

It must be borne in mind that the Law of Dulong and Petit is a law 
on precisely the same footing as Avogadro's Law. According to 
Avogadro's principle, we take as molecular weights those quantities 
of gases which occupy a certain volume under given conditions, and 
from these molecular weights we deduce by a consistent process the 
atomic weights of the elements. According to Dulong and Petit's rule, 
we take as the atomic weights of the solid elements those quantities 
which have a certain capacity for heat. As it happens, the atomic 
weights deduced by the one method coincide with the atomic weights 
deduced by the other. Since, then, the two rules are not contradictory 
but lead to the same result, considerable confidence may be placed in 
the system of atomic weights arrived at by their aid, and the periodic 
regularities referred to in Chapter VI. seem further to justify this trust. 




The rule of Dulong and Petit may be stated in the form that the 
capacity of atoms for heat is approximately the same for all solid 
elements. The atomic heat, as it is called, is obtained by multiplying 
the specific heat of the element into its atomic weight, the product 
being nearly equal to 6*4 when we measure specific heats and atomic 
weights in the customary units. The following table gives the atomic 
weights and the specific heats of the solid elements, and also the 
corresponding products, the atomic heats : — 

Table I 


Atomic Weight. 

Specific Heat. 

Atomic Heat. 

Lithium . 




Beryllium . 




Boron (amorphous) 




Carbon (diamond) 








Magnesium . 




Aluminium . 




Silicon (crystalline) 




Phosphorus (yellow) 




Sulphur (rhombic) 




















Manganese . 








Cobalt .... 




Nickel . 




Copper . 








Gallium . 




Arsenic (crystalline) 




Selenium (crystalline) . 




Bromine (solid) 
























Silver . 








Indium . 
















Iodine . 








Cerium . 












Iridium . 








Gold . 




Mercury (solid) 














Atomic Weight. 

Specific Heat. 

Atomic Heat. 














Lead .... 




It will be observed that though the values of the atomic heats 
fluctuate to some extent, no one diverges greatly from the mean 6*4. 
The diagram on p. 38 shows the regularity between atomic weights 
and specific heats in graphic form, and will be discussed later. 

If we consider for a moment the elements whose atomic heats are 
quite exceptional and give values widely divergent from the mean 6*4, 
we find that they are all elements with low atomic weights. The 
chief are beryllium, boron, carbon, silicon, aluminium, and sulphur. 
Even in these cases the divergence is perhaps only apparent, and that 
for the following reason. The specific heats of all substances vary 
with the temperature at which they are measured ; and though the 
variation is often slight, it is occasionally of relatively great dimensions. 
When this is so in the case of an element, the question arises : At 
what temperature must the measurement of the specific heat be made 
in order to get numbers comparable with those for the other elements ? 
No definite answer has been given to this question, but it is found 
that as the temperature rises, the specific heat seems to approach a 
limiting value, and this value is not in general far removed from that 
which would make the atomic heat approximately equal to 6*4. The 
following tables give the results obtained by Weber for carbon and 
silicon, and by Nilson and Pettersson for beryllium. 








re. Specific Heat. Atomic Heat. 


Specific Heat. 

Atomic I 







+ 10 



+ 10 




















* 5-35 









5 '5 


' 0-4670 











+ 57 















As may be seen in the case of carbon, when an element exists in 
more than one modification, each modification has its own specific heat. 
Thus, at the ordinary temperature the specific heats of diamond and 


graphite are very different. At high temperatures, however, they 
become identical, reaching the same limiting value. 

An approximate rule for compounds of like nature to Dulong and 
Petit's principle for elements was stated by Neumann in 1831. On 
comparing compounds of similar chemical character, he found that the 
products of their specific heats and molecular weights were constant. 
This may be stated in the form that the molecular heats of similarly 
constituted compounds in the solid state are equal. This rule is a 
special case of a regularity of wider scope. Kopp showed that very 
frequently the molecular heat of a solid compound was equal to the 
sum of the atomic heats of its component atoms. The atoms in com- 
bining together to form molecules thus retain their heat capacity 
practically unchanged. By making use of this general principle, the 
specific heats of elements in the solid form have been calculated even 
when the elements themselves were not known to exist in the solid 
state. Thus the specific heat of solid oxygen could not be determined 
directly, but by taking the molecular heats of a great number of solid 
oxygen compounds % and subtracting from them the heat capacities of 
the atoms of the other elements present, the remainder (which was 
supposed to be the heat capacity of the oxygen atoms in the compound) 
always gave an atomic heat for oxygen in round numbers equal to 4. 
This value was therefore taken as the atomic heat of oxygen in the 
solid state, whence on dividing by 16 we get 0*25 as the specific heat of 
solid oxygen. 

The foregoing rules apply only to solids. The specific heat of 
liquids usually varies very much with the temperature, but definite 
relations have been discovered in certain classes of compounds. Thus 
Schiff found that all the ethereal salts of the fatty acids examined 
by him had the same specific heat, no matter what their molecular 
weight was, provided that the specific heats were all measured at 
the same temperature. In other groups of liquid substances less 
simple relations exist, but there is usually an approach to regularity 
of some sort. 

The specific heat of gases presents ^different aspect from the 
specific heats of solids or of liquids. If we compress a gas, we warm 
it, although we supply no heat to it; if we allow a gas to expand 
under atmospheric (or other) pressure, it cools, although no heat has 
been abstracted from it. Now specific heat may be defined as the 
number of heat units which must be supplied to 1 gram of a substance 
in order that its temperature may be raised 1 degree centigrade at the 
ordinary temperature. But we can raise the temperature of 1 gram 
of a gas through 1 degree centigrade, without supplying any heat at 
all, by simply compressing it. We should therefore have to say that 
its specific heat under these conditions is zero, as its temperature has 
been raised without any heat being supplied. On the other hand, if 
we allow the gas to expand sufficiently against pressure while heat is 



being supplied to it, its temperature may be kept absolutely constant, 
the heat supplied merely counterbalancing the cooling due to the 
expansion. The specific heat of the gas in this case is infinite, for a 
finite amount of heat has been supplied and has only produced an 
infinitely small rise of temperature. The specific heat of a gas there- 
fore depends on its volume relations. It may have any positive 
specific heat between zero and infinity, or even any negative specific 
heat, according to the manner in which its volume changes during the 
process of heating, f We must always specify, therefore, under what 
conditions we measure the specific heat of a gas. There are two condi- 
tions which are usually looked upon as standard conditions. In the one 
case the volume of the gas is not allowed to change during the change 
of temperature. This gives the specific heat at constant volume. In 
the other case the volume is allowed to change during the change of 
temperature and the gas is permitted to expand or contract under 
a constant pressure. This gives the specific heat at constant 
pressure, and this for practical measurements is the most convenient, 
the gas being allowed to change its volume at the atmospheric 

The relation between these two specific heats is easily determined. 
Suppose we raise the temperature of 1 g. of gas from 0° to 1°, 
maintaining the volume constant ; the quantity of heat required is the 
specific heat of the gas at constant volume. Let the temperature of 
the same gas be raised now from 0° to 1°, the gas being allowed to 
expand against the atmospheric pressure according to Gay-Lussac's 
Law; the quantity of heat required is the specific heat at constant 
pressure. The quantity of heat required in the second is greater than 
in the first. This is owing to the fact that the gas on expanding 
against pressure cools by doing work, and heat must be supplied to 
make up for this cooling. To find the amount of this heat we measure 
the amount of work the gas does in expanding against the atmospheric 
pressure. As we have already seen in Chapter IV., the work done is 
the product of the pressure into the change of volume. It is thus 
better to consider a certain volume of gas than a certain weight, and 
for purposes of comparisons we take the gram-molecular volume (22*4 
litres at 0° and 760 mm.), i.e. the volume occupied by the gram molecule 
of any gas. By Gay-Lussac's Law the expansion between 0° and 1° 

is — of the volume, viz. _^_ = 82 cc. The work is therefore 

82 x 1033, or about 84,700 gram centimetres. Now to supply 1 cal 
of heat, i.e. to heat 1 g. of water from 0° to 1°, we must do 42,350 
gram centimetres of work. The work done, therefore, by the gram- 
molecular volume of gas on expanding is equal to ^^-° = 2 cal. To 

heat the gram molecule of gas from 0° to 1° requires 2 cal. more at 
constant pressure than at constant volume. It is usual to call the 



amount of heat required to raise the temperature of the gram molecule 
1°, the molecular heat, this being the product of the molecular weight 
into the specific heat. The molecular heat of a gas at constant 
pressure is thus 2 cal. greater than the molecular heat at constant 
volume. The following table contains the molecular heats of some of 
the simpler gases : — 

Table III 

Mol. Heat at 

Mol. Heat at 




Constant Pressure 
C P . 

Constant Volume 

Argon .... 



Helium .... 



Mercury .... 



Hydrogen .... 

H 2 




Nitrogen .... 

N 2 




Oxygen .... 

o 2 




Chlorine .... 

Cl 2 




Bromine .... 

Br 2 




Nitric oxide 





Nitrous oxide . 

N 2 




Carbon monoxide 





Carbon dioxide . 

C0 2 




Sulphur dioxide 
Hydrochloric acid 

so 2 








Sulphuretted hydrogen 

H 2 S 




Ammonia .... 

H 3 N 




Methane .... 

H 4 C 




It is a matter of great difficulty to determine the specific heat of a 
gas at constant volume, owing to a comparatively small amount of the 
heat supplied going to heat the gas, the greater part being absorbed 
by the vessel which contains the gas. It is much easier to determine 
the specific heat at constant pressure, for then we can pass large 
quantities of the gas through a worm contained in a calorimeter, and 
measure the amount of heat it gives or takes. The molecular heats 
at constant volume in the above table have been obtained merely 
by subtracting 2 cal. (the value of the work of expansion per gram 
molecule) from the molecular heat at constant pressure. 

The ratio of the specific heats at constant pressure and 
constant volume can be ascertained with great accuracy from a 
determination of the velocity of sound in the gas under consideration. 
The values of the ratios thus obtained by Kundt's method agree closely 
with those of the above table. In the case of mercury vapour, argon, 
and helium, Kundt's method has alone been used in determining the 
ratios of the specific heats. 

From the kinetic theory of gases it follows that if the molecules of 


a gas are monatomic, so that all the heat they receive from without 
goes only to increase their rectilinear velocity, and not to perform 
any kind of change within the molecule, the molecular heat at constant 

pressure will be 5, and at constant volume 3, i.e. the ratio -~ 

will be 1 '66. If a molecule consists of more atoms than one, we 
should expect that a larger quantity of heat would be absorbed for 
the same rise of temperature, a portion of the heat being employed 
within the molecule in changing the relations of the constituent atoms 
to each other. The molecular heat at constant volume would therefore 
be greater than 3, say 3 + m. But the molecular heat at constant 
pressure is obtained by adding 2 to that at constant volume ; 

it is therefore 5 + m, and the ratio is consequently less than the 


theoretical ratio -• 

Accordingly, when the molecule of a gas consists of a single 
atom, we should expect the ratio of the specific heats to be 1*66; 
for here no heat can be expended in rearrangements of atoms within 
the molecule. Where the molecule of a gas consists of a number 
of atoms, we should expect on the other hand that a portion of the 
heat would be expended in effecting new relations between the 
constituent atoms, and would not go entirely towards increasing the 
rectilineal velocity of the molecule as a whole, and therefore that 
the ratio of the specific heats would be less than 1*66. Until recently 
one elementary gas only was known for which the vapour -density 
results led to the conclusion that there was only one atom in the 
molecule. This gas was mercury vapour, and its density corresponded 
to the molecular weight 200, which is identical with the atomic weight. 
Now Kundt and Warburg, from determination of the speed of sound 
in mercury vapour, found that the ratio of its specific heats is 1*66, a 
figure in accordance with the theoretical value deduced for m6natomic 

The two recently -discovered gases, argon and helium, resemble 
mercury in this respect. The ratio of their specific heats is 1*66, and 
we are therefore obliged to conclude, provisionally, at least, that they 
are elementary substances having only one atom in the molecule. If 
they were compound substances they would necessarily contain more 
than one atom in the molecule, and one would expect the ratio to have 
a lower value. It has been held that argon is an allotropic modification 
of nitrogen, differing from ordinary nitrogen as ozone differs from 
oxygen. On this assumption the molecular weight of argon would be 
42, and its formula would be N 3 . But if this were so, we should expect 
the ratio of the specific heats to be less than the ratio for ordinary 
nitrogen, viz. 1-410, instead of being equal to the theoretical value for 


monatomic gases. Until we have direct evidence to the contrary, 
we must look upon argon and helium as being monatomic, and therefore 
elementary gases ; for the ratio of the specific heats is the only evidence 
we have of the character of their molecules, and that pronounces the 
molecule to be monatomic. 



We have learned in what has preceded that there are about 
seventy distinct chemical substances which have resisted all our attempts 
to decompose them into anything simpler. The question now arises : 


















% s 

,-x S > 


e Qa 




Rh In < 
A9 C $SJ 



w Au 





vie Wt 

>ights — 


Pt xx Hg Bi 

K— X 


100 120 

140 160 
Pig. 1. 

240 260 280 

Are our elements in truth undecomposable, or are they merely com- 
pounds which we are unable to decompose ? It is impossible at 
present to give any definite answer to this question, and it may be 
left open without affecting our views on practical chemical problems at 
all. We may be tolerably certain, however, that if our well-known 

chap, vi THE PERIODIC LAW 39 

elements are really compounds, they are all compounds of the same type, 
that type being different from the compounds of the elements among 
themselves. The reason for this statement is not far to seek. We 
have seen that the product of the atomic weight and specific heat of 
solid elements is nearly constant, arid equal to 6*4 if the ordinary 
emits for atomic weight and specific heat are adopted. We may ex- 
press this relation in the form of a curve (Fig. 1). On the horizontal 
ixis are plotted atomic weights, on the vertical axis specific heats. If 
:he product were exactly equal to 6 "4, the curve would be the smooth 
rectangular hyperbola shown in the figure. The actual values are 
indicated near the symbols, and it will be seen that they never 
tall very far away from the curve. Here, then, we have a regularity 
imongst the elements, a regularity, moreover, which is not shared by 
the compounds (cp. Neumann's Law, p. 33). It is evident, then, from 
this alone, that whatever the ultimate nature of our elements may be 
they form a class of substances apart from all others. 

The same conclusion is reached when we tabulate properties of the 
elements other than the specific heat. Let us take, for example, the 
specific gravity of the elements, and as before tabulate the values on 
the vertical axis, atomic weights being tabulated on the horizontal axis. 
The points thus obtained do not lie, as with the specific heat, approxi- 
mately on a continuous curve, but, as the lines in Fig. 2 indicate, on a 
series of somewhat irregular curves, which resemble each other closely 
in general appearance. If the whole series is looked upon as a single 
curve, that curve is what is termed a periodic curve, i.e. one which 
after a certain interval repeats itself. Each repetition is called a 
period, and the periods have been marked in the figure by Roman 
numerals. The elements alone fall naturally into their places on this 
curve ; compounds, if we take their formula weight as representing 
the atomic weight, fall far away from it, so that we are again justified 
in assuming that our elements, if they are composite, are not com- 
pounds in the ordinary sense. 

The periodic character of the curve shows the existence of many 
relations which do not appear in the curve of specific heats. There 
are evidently two sorts of periods in the figure, the first two being 
very much smaller than the others. These are called the short 
periods ; the others are long periods. We see at once from the curve 
that elements which occupy similar positions in the separate periods 
are similar in their chemical character. Thus the first element in each 
complete period, long or short, is an alkali metal, the second is a 
metal of the alkaline earths, and the last element a halogen. 

It will be noticed, too, that similar elements very often in this 
diagram lie on or near straight lines. Thus if we joint the points for 
lithium and caesium, the first and the last alkali metals, we find that 
the straight line passes close by the points for sodium, potassium, and 
rubidium, the three intermediate alkali metals. Similarly, the points 

sdfrtnvjQ oifmdg 

chap, vi THE PERIODIC LAW 41 

for the metals of the alkaline earths, together with those of the closely- 
related metals beryllium and magnesium, lie very nearly in one 
straight line. The halogens, chlorine, bromine, and iodine, may 
also be connected by a straight line, and so too may arsenic, anti- 
mony, and bismuth ; zinc, cadmium, and mercury ; sulphur, selenium, 
tellurium. The platinum metals (osmium, iridium, platinum ; ruthen- 
ium, rhodium, palladium) lie at the summits of the last complete long 
periods, the related metals iron, nickel, and cobalt occupying the 
corresponding position in the first long period. These relations have 
been indicated in the diagram by means of dotted lines. 

As is the case with the specific heat, it is here again somewhat 
difficult to select the values of the specific gravity which are to be used 
for comparative purposes in such a table as the above. The specific 
gravity varies with the temperature and with the chemical and 
physical states of the substances considered. It is obviously out of 
the question to compare the specific gravity of a gas with that of a 
solid or liquid, so that in the case of elements which are gaseous at 
the ordinary temperature we encounter the chief difficulty. It is true 
that all the well-known elementary gases have been condensed to liquids, 
but the specific gravity of these liquids varies very greatly with the 
temperature. Thus in the case of oxygen the specific gravity at - 130° 
is 0*76, at - 140° it is 0*88, and at the boiling point under atmospheric 
pressure, viz. - 181°, it is 1*12. A decision in the case of nitrogen 
is equally impossible, as Wroblewski gives the following numbers : — 


Pressure in Atmospheres. 

Specific Gravity. 













Here within a range of 60° the specific gravity is doubled. If a 
comparison between the specific gravities of such liquids is to be 
effected at all, the value at the boiling point of the liquid is perhaps 
the most reasonable one to choose. This would give us 1*12 for 
oxygen and 0*83 for nitrogen. For fluorine we have approximately 
1*2, for chlorine 1*35, and for hydrogen, the lightest of all liquids, 0'07. 
The specific gravity of liquid argon at its boiling point is supposed to 
be about 1*5. 

When we come to deal with solid elements the selection between 
different modifications presents itself. Carbon in the form of diamond 
has the specific gravity 3 '3, whilst in the form of graphite it has only 
the value 2*15. Similar differences are found with the various 
modifications of sulphur and phosphorus. The metals, too, according 
to their mode of preparation and physical treatment, have often very 
different specific gravities, that of cobalt, for instance, varying between 
8 '2 and 9*5. In the diagram the highest value has been entered in such 
cases as probably being the most appropriate for comparative purposes. 


There is a function of the specific gravity which is very frequently 
tabulated for the purpose of exhibiting the periodic character of the 
properties of the elements, namely the atomic volume. The atomic 
volume is the product of the atomic weight into the specific volume, 
or, what is the same thing, the quotient of the atomic weight by the 
specific gravity. The curve, first drawn by Lothar Meyer, is shown 
in Fig. 3. The periodic character of the curve is again well marked, 
only now the shape of the periods is different, the maxima corre- 
sponding to the minima of the specific-gravity curve, and vice versa. 
The difference between the short and long periods is once more 
evident, and similar elements once more occupy corresponding positions 
on the curve. In general, we may say that elements on steep portions 
of the curve have very pronounced chemical characters : as we proceed 
from left to right, the elements on descending portions of the curve 
are base forming, and those on ascending portions acid forming. 
Elements at or before the minima cannot be said to be decidedly acid 
forming or decidedly base forming; in different stages of oxidation 
they may be either. 

Most well-defined properties of the elements are periodic, e.g. 
melting point and magnetic power. "When the numerical values are 
tabulated against the atomic weights, the curve obtained is broken up 
into periods, the shape of the period varying according to the property 
tabulated. The regular non-periodic curve obtained for the specific 
heat is quite exceptional. Not only are the properties of the elements 
themselves periodic, but the properties of their corresponding com- 
pounds are also frequently periodic. Thus if we tabulate the heat of 
formation of the chlorides of the elements against the atomic weights 
of the elements which combine with the chlorine, a periodic curve 
results. The same periodicity is observed in the molecular volumes 
of the oxides, and in the melting and boiling points of various corre- 
sponding compounds. 

We may tabulate the elements in another way without reference to 
any particular property such as the specific gravity, but in a manner 
which brings out the general periodic character of the properties of 
the elements with reference to the atomic weights. Newlands 
observed in 1864 that if the elements were arranged in the order of 
their atomic weights, similar elements occurred in the series at 
approximately equal intervals. He stated this regularity in the form 
of a "Law of Octaves," according to which every eighth element in 
the series belonged to a natural group, all the members of which 
resembled each other more than they did the other elements. Thus 
the alkali metals fell into one of these groups, the metals of the 
alkaline earths into another, etc. This rule, however, is not very 
accurate, and had to be relinquished when better data for the atomic 
weights accumulated. It is to Mendeleeff and to Lothar Meyer that 
we owe the periodic arrangement of the elements in its present form. 

sdUfnjo/[ oiuioiy 



In the following table the elements are ordered according to their 
atomic weights, beginning at the top of the left-hand column, pro- 
ceeding downwards to the bottom, then passing to the top of the next 
column on the right, and so on. The tabulation is practically the 
same as that given by Mendeleeff, the only difference being that in 
Mendeleeff s table the elements from lithium to fluorine are placed in 
the same vertical column as the elements from sodium to chlorine, 
so that oxygen and fluorine fall into the same horizontal rows as 
chromium and manganese, whilst sodium and magnesium are placed in 
the same rows as copper and zinc — an arrangement which scarcely 
agrees so well with the general chemical nature of the elements as 
the one which is here adopted. 












2. , 'Be 










... i 





Th j 

S Even Series 


















: Fe 


Os j 



j Co 


Ir | 

" Elements. 


: Ni 


Pt j 


\ Cu 





j Zn 



3. J4$'- 

-•.. Al 

j Ga 


Tl : 

4. JC 


: Ge 


Pb j 

S Odd Series. 

5. J^ 




6. >J 



Te ?""' 


7 *^ F 





K/ I. " 






The vertical columns represent periods, which are denoted by the 
same Eoman numerals as in Figs. 2 and 3. A complete short period 
contains seven elements — a complete long period seventeen, which 
are made up of two series of seven elements each and a transition 


group of three elements. Within any one period, whether short or 
long, there is no sudden change in general chemical properties as we 
pass from one element to the next in order. Descending any series, 
we proceed from base -forming elements, through elements whose 
oxides are neither strongly acidic nor strongly basic, to acid-forming 
elements. . In the short periods, which consist of only one series, we 
start from elements which form powerful bases, and end with elements 
which form powerful acids. In the long periods the uppermost or 
even series start with strong base -forming elements, and end with 
elements which form both acidic and basic oxides. The lower or odd 1 
series, on the other hand, begin with only moderately strong base-form- 
ing elements, and end with elements which are very markedly acid 
forming. The elements connecting the odd and even series in the 
long periods are intermediate in their behaviour between the elements 
they connect; those of the first row (iron, ruthenium, osmium) 
forming acids as well as bases, the others forming only bases. 

When we pass from one period to the next, there is a sudden 
change in the properties of consecutive elements. Thus the last 
element of Period L, F= 19, is in complete contrast in its chemical 
nature to the next element, Na = 23, which belongs to Period II. The 
same thing occurs in the long periods. Iodine, the last member of 
Period IV., forms powerful acids ; while caesium, the first member of 
Period V., is one of the most powerful base-forming elements we 
know. These sudden changes in the chemical properties correspond 
in Figs. 2 and 3 to sudden changes in the direction of the curves. 

If we consider the elements in a horizontal row, denoted by an 
Arabic numeral in the table, we find that they are on the whole such 
as would naturally fall together in a classification of the elements 
according to their general chemical characters. Thus in the first 
row we have lithium, sodium, potassium, rubidium, and caesium, 
metals of the alkalies ; in the second row, magnesium, calcium, 
strontium, and barium, metals of the alkaline earths ; and so on. It 
will be observed that the numbers of the rows in the table are 
repeated, running in order from 1 to 8, and then beginning again at 
1. There are therefore two rows, one in the even and one in the odd 
series, which are represented by the same number. Between the 
members of these two rows there is a general similarity of properties, 
although it is not so great as that between members of the same 
row. Thus the compounds of vanadium, niobium, etc., in the even 
series, bear considerable resemblance to those of phosphorus, arsenic, 
bismuth, etc., in the odd series. The most striking property which 
exemplifies this is the combining capacity or valency of the elements. 
All the elements of two rows indicated by the same number, form, 
in general, compounds with precisely similar formulae. Take, for 

1 The terms " odd " and "even " refer to the numbers of the series in the original arrange- 
ment of Mendeleeff. 


example, the "highest" oxides, i.e. the oxides containing most 
oxygen, of the rows marked 5, which form the two subdivisions of 
one natural family. AVe find their formulae to be : — 

n - f Even series V 2 5 Nb 2 5 - Ta 2 5 

l*roup 5 . • -^odd series N 2 5 P 2 5 As 2 6 5 Sb 2 5 - Bi 2 5 . 

Two atoms of all the elements of Group 5, then, combine with five 
atoms of oxygen. The same thing holds good for other groups, each 
group having its own combining capacity, which increases regularly 
from Group 1 to Group 8, as may be seen in Table II., where instead 
of the elements of Table I. the oxides of these elements have been 
tabulated. In the case of elements forming more than one oxide, the 
highest base-forming or acid-forming oxide is usually taken. The 
oxides enclosed in brackets are not known, but are the anhydrides of 
well-characterised series of salts. 


Th 2 4 










Na 2 

K 2 

Rb 2 



Be 2 2 

Mg 2 2 

Ca 2 2 

Sr 2 2 

Ba 2 2 


Sc 2 o 3 


La20 3 

Yb 2 b 3 


Ti 2 4 

Zr 2 4 

Ce 2 4 



Nb 2 6 




Cr 2 6 

Mo 2 6 

w 2 o 6 


Mn 2 7 
f(Fe 2 6 ) 

Ru 2 8 



i Co 2 3 

Rh 2 4 
Pd 2 4 

lr 2 4 
Pt 2 4 


Cu 2 

Ag 2 
Cd 2 2 

Au 2 


Zn 2 2 

Hg 2 2 


B 2 3 


Ga 2 3 

ln 2 3 

T1 2 3 


C 2 4 

Si 2 4 

Ge 2 4 

Sn 2 4 

Pb 2 4 


N 2 5 

p 2 o 5 

As 2 5 

Sb 2 5 

Bi 2 5 



(Se 2 6 ) 

Te 2 6 



I 2 7 








* The hydroxides corresponding to the monoxides of the alkali metals are well 
known. The existence of many of the oxides themselves is doubtful. 

In order that the amounts of oxygen in the various oxides may be 
readily compared, the formulae have been written throughout so as to 
contain two atoms of the element combined with oxygen, and are 
consequently not in every case the usual formulae. The number of 
the group, with few exceptions, corresponds to the number of oxygen 
atoms united to two atoms of the elements of the group. It must be 
conceded, however, that in order to get this regularity the oxides 
tabulated have been selected somewhat arbitrarily. Thus for copper 
we have selected the lower oxide CugO instead of the equally 
characteristic higher oxide CuO. The selection in the case of 
sulphur, too, is quite arbitrary. Sulphuric anhydride, S0 3 , is not 
the highest acid-forming oxide, as we have the oxide S 2 7 corre- 


sponding to the well-characterised persulphates. Ni 2 3 , Pb0 2 , and 
Bi 2 5 h/.ve no stable acids or salts corresponding to them, and behave 
in general like peroxides. Thus, while there is undoubted periodic 
regularity in the formulas of the oxides of the elements, the regu- 
larit}- is not so definite as at first sight appears from a study of a 
table such as that given above. 

The same sort of regularity appears when we tabulate the highest 
chlorides, bromides, etc., instead of the oxides ; only in these cases 
there are still more exceptions. The highest chlorides of the elements 
of the even series are given in the following table as examples : — 


ThCl 4 
UC1 4 

We here perceive that there is a general correspondence between 
the group number and the number of chlorine atoms with which one 
atom of the elements can combine, but that there is a tendency of the 
elements in the higher groups to unite with fewer atoms than are 
given by the group number. 

When we study the compounds of the elements with hydrogen or 
the alcohol radicals, we find that the combining capacity of the 
different groups is somewhat different from before. In the first 
place, it is chiefly elements of the odd series that unite with hydrogen 
or the alcohol radicals to form definite compounds. As we proceed 
downwards in the periods, we find that the combining capacity does 
not now steadily increase, but reaches a maximum at Group 4, there- 
after to decrease regularly. 

Thus in the first period we have : — 

Table III 










KCl RbCl 



BeCl 2 

MgCl 2 

CaCl 2 SrCl 2 

BaCl 2 


ScCl 3 YtCl 3 

LaCl 3 



TiCl 4 ZrCl 4 

CeCl 3 


VC1 4 NbCl 5 

Ta*Cl 5 


CrCl 3 MoCl 5 

WC1 6 


MnC'l 4 ? 


Methyl Compound. 

Hydrogen Compound 


Li(CH 3 ) 


Be(CH s ) 2 


B(CH 3 ) 3 

BH 3 


C(CH 3 ) 4 

CH 4 


K(CH 3 ) 3 

NH 3 


0'CH 3 ) 2 

OH 2 


F(CH 3 ) 


The power of combining with hydroxyl groups seems to be deter- 
mined by the number of hydrogen atoms with which the element can 
unite. This may be seen from the following table, which gives for 
the elements of the second period those compounds which contain 
most hydroxyl groups : — 






Sodium hydroxide 



Magnesium hydroxide 

Mg(OH) 2 


Aluminium hydroxide 

Al(OH) 3 


Silicic acid 

Si(OH) 4 


Orthophosphoric acid 

PO(OH) 3 


Sulphuric acid 

S0 2 (OH) 2 


Perchloric acid 

C10 3 (OH) 

Corresponding to the oxide P 2 5 there should be the hydroxide 
P(OH) 5 , if for all the oxygen the equivalent amount of hydroxyl were 
introduced. But this compound does not exist, the acid PO(OH) 3 , 
orthophosphoric acid, being the compound with the greatest number 
of hydroxyl groups in the molecule, apparently in connection with the 
fact that there is a hydrogen compound PH 3 , but no hydrogen 
compound PH 5 . 

It has been pointed out that not only are elements in the same 
horizontal row similar, but that there is no sudden change in the 
properties of consecutive elements in the vertical series. We should 
therefore expect to find a general resemblance amongst the elements 
gathered together in one part of the table, and this resemblance does 
in fact appear. The upper rectangle formed by the dotted lines in 
Table I. contains the metals of the rare earths, which occur in groups 
in a few minerals found only in certain localities, and which, from their 
similarity of chemical properties, are extremely difficult to separate 
from one another. The lower rectangle contains all the " metallurgical " 
elements, i.e. the heavy metals which are prepared on the large scale 
from their ores, although the elements enclosed within this rectangle 
are not all of technical importance. Bordering on the rectangle are 
the light metal aluminium and the half -metals arsenic, antimony, and 
bismuth, all of which have commercial importance. 

When we look at the position of the elements composing these 
groups on the curve of specific gravities, we find that the metals of the 
rare earths occupy the middle of the ascending portions of the curve 
in the long periods, and that the " metallurgical " metals lie at or close 
after the maxima. On the curve of atomic volumes, the metals of the 
rare earths are found at the middle of the descending portions of the 
curve, and the metallurgical metals at, or immediately after, the minima. 

The distinction between metals and non-metals is not a very sharp 
one, for the properties of the one class gradually merge into those of 
the other. There is, however, a classification which is generally adopted 
in practice, and this classification is in conformity with the periodic 
table. The elements enclosed in the dotted triangle at the lower left- 
hand corner of Table I. are the non-metals as usually understood. All 
the other elements tabulated are metals. 

As to the blanks in Table L, we may reasonably expect that some 
of them at least will be filled up by elements hitherto undiscovered, 
and that the new elements will be of the same kind as those we already 


know. When the table was first constructed, the number of blanks 
in it was greater than now, and Mendeleeff was bold enough to 
predict the existence of elements to fill the gaps. His predictions 
have been more than once fulfilled. The three well-defined elements, 
gallium, germanium, and scandium, have all been discovered since his 
formulation of the Periodic Law, and have fallen naturally into their 
places in the third period. Mendeleeff not only predicted the existence 
of these elements, but also their chief chemical and physical properties 
from a consideration of the properties of their nearest neighbours in 
the table. By consulting the specific-gravity curve (p. 40), it is easy 
to see for instance, that an element of atomic weight 70 will probably 
have a specific gravity intermediate between the specific gravities of 
zinc and arsenic, the nearest well-known elements on either side. The 
value will therefore lie between 5*7 and 7*2, say about 6 -4. The 
actual specific gravity of gallium, at. wt. 69*7, is 5*93. Similarly the 
melting point and general chemical properties might be roughly pre- 
dicted. It should be said that experiments on the newly-discovered 
elements entirely justified Mendeleeff's prediction of their properties. 

Another use to which the periodic table has been put is the correc- 
tion of supposed erroneous atomic weights. For instance, the atomic 
weights of the elements of Group 8 presented in some cases exceptions 
to the Periodic Law (the elements when arranged in order of their 
atomic weights being out of harmony with the natural grouping), and 
were therefore subjected to a revision, the result of which has been to 
show that the requirements of the Periodic Law are in reality fulfilled. 
Again, the atomic weight of tellurium was found to be greater than 
that of iodine, which caused iodine to come before tellurium in the 
periodic classification, so as to fall into the sulphur group, while 
tellurium fell into the halogen group. But this contradicts the whole 
chemical behaviour of these elements ; iodine from its properties must 
be classed along with chlorine and bromine, and tellurium naturally 
belongs to the sulphur and selenium group. It was therefore thought 
probable that one or other of the atomic weights had been incorrectly 
^determined, doubtless that of tellurium, as iodine has been the subject 
of repeated concordant determinations. Fresh investigations at first 
seemed to show that tellurium had an atomic weight of about 125, in 
'accordance with the Periodic Law, but the most recent work has 
pointed to a value of slightly more than 127, or a fraction above the 
[atomic weight of iodine. We have here then either a well-marked 
1 3xception to the periodic regularity, or else tellurium and its compounds 
have never been obtained in the pure state. The latter alternative is 
perhaps the more probable, but until further careful work has been 
Hone on the subject no proper decision can be arrived at. 

The position of the new gases, helium, argon, etc., in the periodic 
'stable is still under discussion, the data being too scanty to allow any 
final conclusion to be drawn. 



The solutions we most frequently meet with in inorganic chemistry 
are solutions in which water is the solvent. The substances dissolved 
may be gaseous (e.g. ammonia, hydrochloric acid) or liquid (e.g. 
bromine), but in the vast majority of cases they are solid. 

When excess of a solid, such as common salt, is brought into 
contact with water at a given temperature, it dissolves until the 
solution reaches a certain concentration, after which the solution may 
be left for an indefinite time in contact with the solid without either 
undergoing any change, provided the temperature is kept constant, 
and that evaporation is prevented. Such a solution is said to be 
saturated with respect to the salt at the given temperature, and the 
solubility of salt in water under these conditions is merely an expression 
for the strength of the saturated solution. The strength may be given 
in various ways, e.g. 100 parts of water are said to dissolve at the given 
temperature so many parts of salt, or the saturated solution contains so 
much per cent of the solid, or so many grams of salt are contained in 
100 cc. of the solution. All these methods of expressing the strength of 
solutions are in common use, the last especially for volumetric work, 
and it is a matter of convenience which to choose in any particular 
case. When a solution is capable of dissolving more salt at the given 
temperature than it already contains, it is said to be unsaturated ; 
when it is made by some device to contain more salt than it would 
be capable of taking up directly under the given circumstances, it] 
is said to be supersaturated. The one and only test to find whether 
a solution is saturated, unsaturated, or supersaturated, with respect 
to a given solid, is to bring it in contact with that solid. If it is 
saturated, it will remain unchanged; if unsaturated, it will dissolve 
more solid ; if supersaturated, it will deposit the excess of dissolved sub- 
stance till the strength diminishes to that of the saturated solution. 

The solubility of all substances varies with the temperature. 
With most solids it increases as the temperature rises. Thus 100 
g. of water will dissolve 35*6 g. of sodium chloride at 0°, and 




40 g. at -100°; 47 g. of sodium thiosulphate x at 0°, and 102 g. 
at 60°; 13 g. of potassium nitrate at 0°, and 247 g. at 100°. 
In some cases, however, the solubility diminishes as the temperature 
rises. For example, calcium citrate is more soluble in cold water 
than in boiling water, and many other calcium salts of organic 
acids exhibit a similar behaviour. The change of solubility with 
change of temperature may be seen in the accompanying diagram, 
the solubilities being given as parts of anhydrous salt dissolved in 
100 parts of water. The curves in general are continuous, but the 
curve representing the solubility of sodium sulphate is not continuous, 
having a sudden break at 33°. The explanation is that we are not 



here dealing with one solubility curve but with two solubility curves. 
The curve below 33° is the solubility curve of the hydrate Na 2 S0 4 . 
10H 2 O ; the curve above 33° is the solubility curve of the anhydrous 
salt Na 2 S0 4 . When the hydrate is heated to 33° it splits up into 
water and the anhydrous salt. The hydrate, therefore, has no stable 
existence above 33°, and the only solid with which the solution can 
be in contact above that temperature is the anhydrous salt, and the 
curve therefore represents its solubility and not that of the hydrate. 
That we are here actually dealing with two curves may be proved by 
bringing the anhydrous salt in contact with water below 33°. It 

1 Calculated as anhydrous salt. 


does not at once unite with water to form the solid hydrate, so that 
we may measure its solubility at such temperatures and represent it 
by means of the dotted line in the figure. It will be seen that the 
dotted line is a continuation of the upper curve, and is quite 
independent of the curve of the hydrate, except in so far as it cuts 
it at the temperature of the transformation of the hydrate into the 
anhydrous salt. If we take a point in the diagram lying between 
the dotted line and the lower curve, that marked a, for example, we 
find that it represents the strength of a solution which is unsaturated 
with respect to the anhydrous salt, and supersaturated with respect to 
the hydrate Na 2 S0 4 . 1 0H 2 O. We can prove that this is so in reality, 
for if we bring the solution into contact with the anhydrous sulphate 
it will dissolve some of the salt ; whilst if we bring it into contact 
with the hydrated salt it will deposit a fresh quantity of the hydrate. 
It is very necessary, then, to specify exactly the solid with respect to 
which a solution is said to be saturated or not. To speak of a 
solution being a saturated solution of sodium sulphate is ambiguous, 
for while the solution may be saturated with regard to one form of 
the solid salt, it may be unsaturated or supersaturated with regard to 

When a saturated solution of hydrated sodium sulphate, Na 2 S0 4 . 
10H 2 O, is made in warm water so that none of the solid remains, and 
is then cooled to the ordinary temperature, care being taken that 
there is no separation of the salt, the solution may be kept unaltered 
in the supersaturated state for a great length of time. It frequently 
happens that the solution on standing deposits crystals of another 
hydrate, Na 2 S0 4 . 7H 2 0, which at low temperatures is more soluble 
than the decahydrate. The solution then becomes saturated with 
respect to this heptahydrate, but is still supersaturated with respect 
to the decahydrate, as may be seen by dropping in a crystal of the 
latter, when further crystallisation at once commences. It is mostly 
in the case of salts which crystallise with water of crystallisation that 
we can easily form supersaturated solutions. Salts which separate 
only in the anhydrous state do not often form aqueous supersaturated 
solutions, the excess being usually deposited as the cooling progresses. 
Sodium chlorate, however, is a good example of an anhydrous salt 
which does form supersaturated solutions. 

The question of the transformation of different hydrates into each 
other will be discussed in the chapter on the Phase Rule. 

Liquids sometimes mix with each other in all proportions, e.g. water 
and alcohol ; sometimes not at all, e.g. water and mercury. In some 
instances there is partial miscibility. When we shake up water 
and ether together in about equal proportions, the water dissolves a 
little of the ether, and the ether a little of the water, the two saturated 
solutions then separating. The lower layer is a saturated aqueous 
solution of ether, the upper layer is a saturated ethereal solution of 





water. 1 Suppose that with two partially miscible liquids, A and B (say 
aniline and water), the solubility of each liquid in the other increases 
with rise of temperature. On raising the temperature we should find 
that the composition of the two satur- 
ated layers, or phases, as they are 
called, would tend to approximate to 
the same value. We may represent 
this diagrammatically, as in Fig. 5. 
Temperatures are plotted on the vertical 
axis, and points on the horizontal axis 
represent the percentage composition 
of the liquids. 

As the temperature rises, the solu- 
bility of B in A and of A in B in- 
creases, so that the corresponding points 
on the two curves approximate. At 
a certain temperature the two curves 
will meet. This means that at that 
particular temperature the solution of 

A in B and the solution of B in A have the same composition, i.e. are 
identical. At this temperature the liquids are miscible in all propor- 
tions. There is therefore no fundamental distinction between wholly 
miscible and partially miscible liquids ; for liquids which only mix 
partially at one temperature may mix in any proportion at another. 
We have here, likewise, an illustration of the arbitrary manner in 
which we employ the terms " solvent " and " dissolved substance." We 
have spoken of the solubility of A in the solvent B for one part of 
the curve, and the solubility of B in A for the other. At the 
temperature where the saturated solutions become identical it is 
obviously impossible to make any distinction between solvent and 
dissolved substance ; and in general, it may be said that, though for 
the sake of convenience we often thus discriminate between the two 
constituents of the solution, there is in the solution itself no actual 
distinction of this kind. If it suits our purpose we may call dissolved 


Per cent Aniline 
Fig. 5. 

The following table gives the mutual solubility at 22° of water and some common 

organic solvents : — 


Substance in 

Vols. Water in 

100 Vols. Water. 

100 Vols. Substance 







Carbon bisulphide 









Amyl alcohol 






It has been been pointed out that, as far at least as water and organic liquids are con- 
cerned, partial miscibility involves the presence of an " associated" liquid (cp. Chapter 
XIX.), "non-associated" liquids being completely miscible. 



substance what is usually termed solvent, and vice versa, without fear 
of committing any theoretical error. 

The solubility of some pairs of miscible liquids in each other 
increases with rise of temperature, e.g. aniline and water (Fig. 5) ; 

the mutual solubility of other pairs 
diminishes, e.g. dimethylamine and 
water (Fig. 6). The diminution of 
solubility by rise of temperature is 
very easily seen in the case of water 
and paraldehyde. If these liquids 
are shaken up together in a test-tube 
at the ordinary temperature, they 
separate on standing into two clear 
layers. If now the test-tube is 
plunged into warm water, the 
aqueous layer at once becomes 
turbid from separation of the excess 
ioo of paraldehyde. 

Many lactones exhibit a peculiar 
behaviour with water (Fig. 7). 
The solution prepared by shaking up the two substances at the 
ordinary temperature becomes turbid at about 40° owing to the 
lessened mutual solubility, but at 80° again becomes clear. Here 

Per cent Dimethylamine 

Fig. 6. 

Per cent Lactone 

Fig. 7. 

Per cent 

Fig. 8. 

there is first diminution of the solubility of the lactone in water by 
rise of temperature to a minimum value, after which we have 
increasing solubility with further rise of temperature. It is con- 
ceivable that the solubility curve of some particular lactone might be 
a closed curve, as in Fig. 8, there being complete miscibility at high 
and at low temperatures, whilst at intermediate temperatures there 
would be only partial miscibility. 


Gases vary very much with respect to their solubility in liquids. 
For example, one volume of water at 0° and 760 mm. dissolves — 

1050 vols. Ammonia. 


r Hydrochloric acid. 


, Sulphuretted hydrogen, 


„ Sulphur dioxide. 
,, Carbon dioxide. 



, Ethylene. 


, Oxygen. 


, Argon. 

0*02 , 

, Nitrogen. 


, Hydrogen. 

Again, one volume of alcohol under the same conditions dissolves — 
17*9 vols. Sulphuretted hydrogen. 

4'3 , 

, Carbon dioxide 

3*6 , 
0*28 , 
0*13 , 
0-07 , 

, Ethylene. 
, Oxygen. 
, Nitrogen. 

Other conditions being the same, the quantity of a gas dissolved by 
a liquid falls off with rise of temperature. 

When the gases are only moderately soluble in the solvent 
chosen, they obey a simple law with regard to pressure known as 
Henry's Law. One mode of stating this law is : A given quantity of 
the liquid solvent will always dissolve the same volume of a given gas, 
no matter what the pressure may be, so long as the temperature is 
constant. Another equivalent form is : A given quantity of the 
liquid will dissolve at constant temperature quantities (by weig ht') of 
-the gas which are proportional to the pressure of the gas. I'nus, if a 
quantity of water will dissolve 1 g. of oxygen at the atmospheric 
pressure, it will dissolve 2 g. of oxygen at a pressure of 2 atm. 
The equivalence of the two forms of the law may of course at once be 
seen by considering that doubling the pressure of a gas halves its 
volume, so that the 2 g. of oxygen at a pressure of 2 atm. occupy the 
same volume as 1 g. of oxygen at a pressure of 1 atm. 

When a mixture of two or more gases is dealt with, each dissolves 
in the liquid as if all the others were absent. This rule may be also 
stated in the form, that when a mixture of gases dissolves in a liquid, 
each component dissolves according to its own partial pressure, and in 
this form it is called Dalton's Law. These two laws only hold good 
with accuracy when the gases are comparatively slightly soluble in 
water, and when the pressures do not exceed a few atmospheres. 
With very soluble gases and at great pressures there are great 
divergencies, at least from Henry's Law. This is probably owing in 
many cases to chemical or quasi-chemical changes which take place 
when the gas dissolves in water. The gases which dissolve freely in 


water are all of a more or less pronouncedly acidic or basic nature. 
Thus amongst the most soluble gases are ammonia and the hydrogen 
compounds of the halogens. Neutral gases, which do not by union 
with water form acidic or basic substances, are only sparingly soluble in 
water, e.g. the chief gases of the atmosphere, the gaseous hydrocarbons, 
etc. There is also a general connection between easy compressibility 
to the liquid state at atmospheric temperatures and easy solubility in 
water. The so-called "permanent gases," nitrogen, oxygen, argon, 
helium, carbon monoxide, nitric oxide, methane, are all very slightly 
soluble. The simple gases which are easily condensed to liquids are 
much more soluble. 

In general, we may state that when a substance exhibits chemical 
relationship with a liquid it is more or less soluble in that liquid. 
Compounds containing hydroxyl in organic chemistry are very 
frequently soluble in water, and the more hydroxyl they contain 
proportionally in the molecule the more soluble they are. Thus the 
higher monohydric alcohols of the series CJB^+iOH, say C 6 H 13 OH, 
are scarcely soluble, whilst the polyhydric alcohols, such as mannitol, 
C 6 H 8 (OH) 6 , are freely soluble. Hydrocarbons are, as a rule, very 
sparingly soluble in simple compounds containing hydroxyl, e.g. water, 
alcohol ; and substances with proportionately much hydroxyl are 
very slightly soluble in hydrocarbons. The solid hydrocarbons are, 
however, freely soluble in the liquid hydrocarbons, and that the 
more as the solvent and dissolved substances approach each other more 
nearly in character. Sulphur and the metals are insoluble in water ; 
the former, however, dissolves freely in liquid sulphur compounds, e.g. 
carbon disulphide and sulphur chloride, whilst the latter are often 
soluble in the liquid metal mercury, or in other metals in the fused 
state, whence the possibility of forming alloys and amalgams. 

The solubility of salts, acids, and bases in water is very variable ; 
but the following rules for ordinary compounds may be of use to 
the student : — 

Salts of potassium, sodium, and ammonium are soluble. 

Normal nitrates, chlorates, and acetates are soluble. 

Normal chlorides are soluble (except those of silver, mercurosum, 
and lead). 

Normal sulphates are soluble (except those of calcium, strontium, 
lead, and barium). 

Hydroxides are insoluble 1 (except those of potassium, sodium, and 
ammonium, which are freely soluble, and those of calcium, strontium, 
and barium, which are sparingly soluble). 

Normal carbonates, phosphates, and sulphides are insoluble 
(except' those of potassium, sodium, and ammonium). 

Basic salts are, as a rule, insoluble. 

1 By "insoluble" we generally mean "very sparingly soluble." The relative solu- 
^ bilities of some common " insoluble " substances are given in Chapter XXVI. 


Acid salts are, as a rule, soluble if the acid is soluble, which is 
mostly the case with inorganic acids. 

A process of purification very frequently made use of is that of 
recrystallisation. This consists in dissolving the impure substance 
and allowing a portion of it to separate out of the solution, either by 
a change of temperature that will lower the solubility, or by evapora- 
tion of the solvent. Suppose the substance to contain 90 per cent of 
pure material and 10 per cent of an equally soluble impurity. The 
mixture is treated with as much solvent as will dissolve the whole. 
If we make the temporary assumption that the substances have no 
influence on each other's solubility, the solution is now saturated with 
respect to the pure material, but far below the saturation point of 
the impurity. By removing part of the solvent, say by evaporation, 
we make the solution supersaturated with regard to the pure material, 
so that a portion of the latter is deposited. The solution still remains 
unsaturated, however, with regard to the impurity, so that none of 
this component falls out. We can therefore go on removing the 
solvent, and so obtain crystals of the pure material, until we reach 
the saturation point of the impurity, when the operation must be 
stopped, for then pure substance and impurity would deposit in equal 
proportions. In this way f-ths of the original substance would be 
obtained in the pure state by the sacrifice of ^-th, which would be 
more highly contaminated with the original impurity. Purification 
by crystallising is always attended by loss of material, but if the 
process of recrystallisation is carried out systematically, this loss 
in the majority of cases need only be slight. In no case is the 
process so simple as in the scheme given above, and it is almost 
always necessary to repeat the crystallisation several times, the separa- 
tion being even then by no means perfect. In organic chemistry 
the success of purification by crystallisation largely depends on the 
choice of solvent, a proper choice leading to a rapid and perfect 
separation with a good yield of pure substance, where an unsuitable 
selection would give a small yield of still impure material. Where, 
as in the case of the Stassfurt salts, complex recrystallisations of 
aqueous solutions are carried out, the temperatures of crystallisation 
are carefully chosen, and the nature of the solvent for any particular 
salt is varied by using solutions of other salts of varying concentration. 
In this way, and in successive operations, it is found possible to obtain 
pure potassium chloride from the very complex salt deposits. 

A method of separation often practised in organic chemistry is 
extraction from aqueous solution by means of ether. The principle 
of the process is as follows. If a substance is soluble in two liquids 
which are not themselves miscible, it will distribute itself between the 
two solvents, when shaken up with them in the same vessel, in a 
manner depending on its solubility in each of the solvents separately. 
The simplest case is when the solvents are perfectly immiscible, and when 


the dissolved substance has the same molecular weight in both. The 

rule for this case is that the substance on shaking up distributes itself 

between the two solvents in such a proportion that the ratio of 

the concentrations of the two solutions is equal to the ratio of the 

concentrations of solutions saturated separately with the substance at 

the temperature of experiment, i.e. the ratio of the concentrations is the 

ratio of the solubilities in the separate solvents. Suppose, for example, 

that the immiscible substances are water and benzene, that the 

molecular weight of the substance in these two liquids is the same, 

and that the substance is twice as soluble in benzene as in an equal 

volume of water. On shaking up the substance with benzene and 

water in quantity insufficient to saturate both these solvents, we shall 

find that the substance distributes itself so that the concentration of 

the benzene solution is twice that of the aqueous solution. It will be 

seen that nothing is here said as to the absolute or relative volumes 

of the solvents — it is entirely a question of final concentrations. Using 

the above rule, we might consider the practical question : Whether 

is it more economical in the above instance to extract the aqueous 

solution with an equal volume of benzene in one operation, or to 

apply the benzene in several portions, shaking up and separating 

each time ? If the amount of substance in one volume of the aqueous 

solution is A, then on shaking up with one volume of benzene, we 

shall have one-third of A remaining in the aqueous solution, and 

two-thirds of A in the. benzene. We thus with one volume of 

benzene extract 0*67^ if we perform the extraction in one operation. 

Let us now suppose the extraction to be performed in two stages, 

using the same total quantity of benzene as before. To one volume of 

the aqueous solution we add half a volume of benzene. Let x be the 

amount extracted by the benzene. A - x will be the amount left in 

one volume of water; x is the amount in 0*5 volume of benzene. 

The concentrations are therefore as A - x to r-~ = 2x. But by the 

above rule the concentration in the benzene is twice that in the water, 

so we have . y 

2x = 2(A - x) 

x = 0'5A. 

We have therefore by using half a volume of benzene extracted 0*5 A, 
leaving OS A behind in the water. Extracting this aqueous solution 
again with half its volume of benzene, we shall again remove half of 
what it contains, viz. 0*25^; so that by extracting in two successive 
operations we obtain 0'5A + 0'25A = 0'75A instead of 0*67^, the 
result obtained by using the same quantity of benzene in one opera- 
tion. If we are therefore desirous of getting the maximum amount 
extracted with a given quantity of the extracting solvent, and time is 
not in consideration, it is better to apply the extracting liquid in suc- 
cessive small portions than to use all in one operation. 


In the actual extraction with ether economy of time has more 
frequently to be considered than economy of ether, and larger quanti- 
ties in fewer operations are as a rule employed. This may the more 
readily be done as the easy volatility enables one portion to be distilled 
off and made ready for a second operation while extraction with an- 
other portion is in progress. The case of ether and water is not the 
ideal case for which the solubility rule applies accurately. Ether and 
water are partially miscible, and the partition coefficient of a substance 
between them is on account of this not the ratio of the solubilities of 
the substance in pure ether and pure water separately. It very 
frequently happens, too, that the molecular weight of the substance in 
water is not the same as the molecular weight of the substance 
dissolved in ether (see Chap. XIX.). There is then no constant 
" partition coefficient " at all, and the ratio of the concentrations of the 
two solutions depends on the relative proportion of dissolved substance 
and solvents present. The effect of a difference in molecular weight 
being exhibited in the two solvents will be discussed in the chapter on 
Molecular Complexity. 

The resemblance between the above problem and that of shaking 
a gas and liquid up together in a closed vessel is evident. If we 
regard "vacuum" as a solvent, we may state Henry's Law in the form 
that there is always a definite "partition coefficient" between the 
vacuum and the liquid solvent, this partition coefficient being the 
solubility of the gas in the liquid. Let equal volumes of water and 
nitrogen be shaken up in a stoppered bottle. The solubility 
of nitrogen is n, i.e. one volume of water dissolves n volumes of 
nitrogen. The final concentration of the gas in the water is thus 
n times the final concentration of the gas in the vacuum ; or the 
partition coefficient between water and vacuum is n : 1 = n. The 
original amount of nitrogen is, say, N. Let xN be the amount dis- 
solved. We have N - xN for the amount remaining, and as the 
volumes of water and vacuum are equal, the final concentrations are 
N-xN said xN respectively ; but these are in the ratio of 1 to n, so 
that we have 

N-xN _l-x_\ 
xN x n 


1 + n 

Since according to Dalton's Law the presence, of one gas does not affect 
the solubility of another, we can easily solve problems concerning the 
solubility of mixtures of gases in the same solvent. The student will 
find it a useful exercise to calculate from the tables of solubilities given 
above, the pressure and composition of the residual gaseous mixture of 
oxygen, nitrogen, and argon obtained by shaking up 1 vol. of purified 
air with 1 vol. water at 0° and 760 mm. in a closed vessel. 



When a solid such as glass is heated, it gradually loses its rigidity, 
and by degrees, as the temperature is raised, assumes the fluid form, 
passing through the stages of a tough, inelastic solid and of a viscid, 
pasty liquid. It cannot be said, therefore, to have a definite melting 
point. A crystalline substance such as sulphur, on the other hand, 
when heated remains solid up to a certain temperature, after which the 
further application of heat liquefies it, the temperature remaining con- 
stant during liquefaction. This constant temperature is the melting 
point of the solid, and is a characteristic property of each particular 
crystalline substance^ Not only is the melting point used as a means 
of identification, but sharpness of melting point, i.e. constancy of tem- 
perature during liquefaction, is often employed as a test of purity of 
crystalline substances, especially in organic chemistry. Should a sub- 
stance crystallise in more than one form, each crystalline variety has 
its own melting point. The melting point of rhombic sulphur, for 
example, is 114*5°, while the melting point of monoclinic sulphur is 

When a fused substance is cooled below the point at which the 
solid melted, it may or may not solidify according to circumstances. 
Once the solidification does begin, however, the temperature adjusts 
itself to the melting point of the solid, for that is the only temperature 
at which the solid and the liquid can permanently exist in contact. Of 
course we may put a piece of ice into water of 15°, and the two may 
coexist for a time though the temperature of the ice is 0°. But here 
there is no permanent coexistence, for the ice is constantly melting, 
and the temperature of the water is being lowered. If .we wish to 
ascertain the exact temperature at which the solid and liquid coexist, 
we must ensure their intimate mixture, otherwise the solid might be at 
one temperature and the bulk of the liquid at another. 

When a crystallised solid is heated, it is apparently impossible to 
warm it above its melting point without its assuming the liquid state. 
It is easy, however, in the case of most liquids to lower their tempera- 


ture below the melting point of the solid without actual solidification 
taking place. The freezing point of a liquid, therefore, if defined as 
the temperature at which the liquid begins to freeze, is not a definite 
temperature, and as a matter of practice only melting points of pure 
substances are determined. Although ice when heated melts at 0°, 
water may be cooled to - 4° or lower without any separation of ice 
taking place. The liquid is then said to be superfused. The intro- 
duction of a crystal of the solid into the superfused liquid at once 
determines crystallisation, so that a superfused liquid in this respect 
resembles a supersaturated solution (p. 50). 

Fusion is always attended by absorption of heat, and solidi- 
fication by evolution of heat. When a superfused liquid begins to 
solidify, therefore, heat is disengaged and the temperature of the liquid 
rises. This rise of temperature accompanying the separation of the 
solid goes on until the melting point of the solid is reached, when no 
further change occurs, for at this temperature the solid and liquid are 
in equilibrium, i.e. can exist together in any proportion without affect- 
ing each other. If there is no heat exchange with the exterior, the 
solid and liquid remain in contact at their melting point without altera- 
tion. If heat is added from an external source, part of the solid lique- 
fies, thereby absorbing the heat supplied. If heat is withdrawn from 
the mixture, part of the liquid solidifies, thereby producing sufficient 
heat to allow the temperature to remain constant at the melting point. 

It has been said above that the introduction of a crystal of the 
substance into a superfused liquid is in all cases sufficient to deter- 
mine its crystallisation. This would appear to be true even when 
the crystalline particle is so minute as to be unrecognisable by 
ordinary means, for experiments have shown that as small a quantity 
as one-hundred-thousandth part of a milligram is effective in producing 
solidification. Experimental researches have also been made to 
investigate the tendency of a superfused liquid to crystallise by 
itself, the possibility of the introduction of any crystallised particle 
from without being excluded. In most superfused liquids crystalline 
nuclei soon make their appearance at different parts of the liquid, 
and then begin to grow until all the liquid has solidified, or until the 
heat disengaged on solidification has raised the temperature of the 
whole to the freezing point. It is obvious that the chance of such a 
nucleus appearing increases with the quantity of liquid taken, and we 
might therefore expect to keep a small portion of a superfused liquid 
for a longer time uncrysfcallised than we should a larger quantity. 
This is found to be in accordance with fact, and it is a comparatively 
easy matter to count the number of nuclei formed in a given time if 
the superfused liquid is enclosed in a capillary tube, and to determine 
by measurement the rate at which these nuclei grow. It has been 
ascertained that the rate of growth is approximately proportional to the 
degree of superfusion when that degree is not very great, and that 


the number of nuclei formed in a given volume in a given time at 
first increases with the degree of superfusion, but afterwards reaches a 
maximum, and begins to diminish as the liquid becomes highly super- 
fused. It is possible, therefore, by suddenly cooling a fused liquid far 
below the proper freezing point, to diminish the tendency to nucleus 
formation so greatly that the substance may be kept for days, 
weeks, or even months, without crystallisation taking place. Such a 
highly superfused substance is no longer a liquid but a solid of the 
same nature as glass. It has no definite melting point, but would, if 
crystallisation were prevented, like glass become gradually softer and 
less viscous until it might be said to have assumed the ordinary 
liquid form, there being at no time a sudden passage from the solid to 
the liquid state. If such a glassy solid is brought into contact with 
the corresponding crystalline substance, it crystallises, but the velocity 
of the crystallisation is extremely small, the rate being measurable 
in millimetres per hour. We have then no sharp line of demarcation 
between amorphous solids and liquids, but a gradual passage from the 
one state to the other. The line of demarcation is rather between 
crystalline substances and non-crystalline substances, i.e. between 
substances which have their particles arranged in a regular manner, 
and those in which the arrangement is confused and irregular. It 
was for long supposed that no regular arrangement of particles could 
subsist in liquids, the particles of which have a certain freedom of 
motion not possessed by solids, but of late crystalline liquids have 
been discovered which possess properties hitherto only encountered in 
solid crystals. Thus when para-azoxyanisoil is heated, it melts at 
114°, but the liquid obtained exhibits a somewhat turbid appearance 
and strong double refraction. This double refraction points to a regular 
arrangement of particles in the substance, which is yet undoubtedly a 
liquid so far as its mechanical properties are concerned. It flows easily, 
and rises in a capillary tube, and from the capillary rise its molecular 
weight may be calculated by the method of Ramsay and Shields 
(Chap. XVIII. ). At 134*1° it suddenly loses its turbidity and double 
refraction, and becomes in all respects similar to an ordinary liquid. 
The transition from the state of " crystalline " liquid to ordinary 
liquid is accompanied by a diminution in density, but by no change in 
the molecular weight. On cooling, the reverse changes occur : the 
ordinary liquid passes into the crystalline liquid at 134"1°, and this 
into the crystalline solid at 114°. 

The phenomena of nucleus formation, nuclear growth, and extreme 
superfusion can be very easily shown with hippuric acid, a substance 
melting at 188°. A small quantity of it is melted at a temperature not 
exceeding 195°, in order to avoid decomposition, and a portion of the 
liquid is sucked up into a thin-walled capillary (melting-point tube) about 
6 inches long and 1 mm. or less in diameter, the lower end "of which 
is drawn out to a fine jet but not sealed. The liquid is manipulated 


above a flame until it settles in the middle of the tube, and the 
narrow end of the capillary is then carefully sealed off. The tube is 
held horizontally by the open end, and moved above the flame until 
every crystalline particle has disappeared, after which it is suddenly 
chilled by dipping into a beaker of cold water. The contents have 
then assumed the state of a transparent glass. If the tube is broken, 
it is found that the acid is undoubtedly solid in the ordinary sense, 
though not crystalline. When left to itself it remains unchanged for 
days, but if it is dipped into water at 100°, nuclei will appear and 
grow along the tube at a moderate rate. The growth may be 
interrupted at any time by chilling the tube in cold water. If a 
lighted taper be passed rapidly beneath the tube, nuclei will be 
found to form at the heated portions and not elsewhere. 

When a foreign substance is dissolved in a liquid, the freezing 
point of the solution is lower than that of the pure solvent. It is, 
strictly speaking, necessary here to specify what substance separates 
out, but unless it is otherwise expressly indicated we shall always 
assume that it is the solidified solvent. A solution of a substance in 
water must be cooled below the freezing point of water before ice 
separates out, and for small concentrations the lowering of the freezing 
point is proportional to the amount of the substance in solution 
(Blagden's Law). The freezing point of a solution is strictly defined 
as the temperature at which the solution is in equilibrium with the solid 
solvent. At this temperature it may be mixed with the solid solvent 
in any proportion without undergoing change ; it neither melts the 
solid nor deposits any solvent in the solid form. Practical everyday 
applications of the lowering of the freezing point by substances in 
solution are to be found in the melting of snow or ice by salt, and in 
the addition of alcohol or glycerine to wet gas-meters to prevent 
freezing in cold weather. In the latter case the alcohol or glycerine 
added to the water lowers its freezing point greatly ; a mixture con- 
taining, for instance, 25 per cent of alcohol freezes at - 13°, so that the 
external temperature must fall far below the freezing point before the 
gas-meter ceases to act. Glycerine has a certain advantage over 
alcohol for this purpose, inasmuch as it is non-volatile. When alcohol 
is used it slowly evaporates, and the solution, owing to this cause, 
becomes both smaller in bulk and weaker. Glycerine not only does 
not evaporate itself, but also greatly reduces the vapour pressure of 
the water (Chap. IX.) so that it both diminishes the chances of freezing 
and prevents rapid evaporation. 

The action of salt in melting snow or ice may be seen from the 
following consideration. A solution containing a little salt is in 
equilibrium with ice at a temperature somewhat below 0°. If 
we cool the solution below this particular temperature, ice will 
separate, and the remaining solution will therefore become more con- 
centrated. Its freezing point will consequently be lower than that of 




the original solution. If we continue the cooling process, more and 
more ice will fall out and the solution will become still more concen- 
trated and have a still lower freezing point. This process might be 
conceived to go on without limit if salt were infinitely soluble in 

water. But we know that 
there is a limit to the solu- 
bility of salt in water, i.e. 
at a given temperature a 
certain quantity of water 
will only take up a definite 
amount of salt and no more. 
This finite solubility of salt 
in water accordingly sets a 
limit to the lowering of the 
freezing point. We may 
best understand this by 
making use of a combined 
solubility and freezing-point 
diagram. The horizontal 
axis gives the composition, 
i.e. percentage of salt in 
the mixture. Temperatures 
are plotted on the vertical 
axis. If we lower the 
temperature of a solution containing a little salt, ice will begin to 
separate out and the freezing point will fall as the concentration of 
the remaining solution increases. The relation between the concentra- 
tion of the solution and its freezing temperature (i.e. the temperature 
at which it is in equilibrium with ice) is represented by the curve 
on the left, which we shall call the freezing-point curve. If on the 
other hand we begin with a saturated solution of salt above 0° and cool 
it, salt will separate out and the solution will become less concentrated, 
for the solubility of the salt diminishes with falling temperature. 
The relation between the concentration and temperature at which the 
solution is in equilibrium with the solid salt is given by the curve 
on the right, which is therefore an ordinary solubility curve. The 
two curves evidently tend to approximate with fall of temperature, 
the concentration of the solution which is in equilibrium with ice 
increasing, and the concentration of the solution which is in equilibrium 
with salt diminishing ; and at a certain temperature they meet. 

We shall now consider what occurs when we follow a salt solution 
down either of the curves to the point C where the curves intersect. 
Starting with a weak solution, we find that as the temperature falls, 
ice separates and the solution becomes more concentrated. This goes 
on until the freezing-point curve cuts the solubility curve, at which 
point the solution remaining behind becomes saturated. If we 


Fig. 9. 


attempt to lower the temperature still farther, ice separates out ; but 
the solution remaining is now supersaturated, and therefore deposits 
salt until the saturation point is again reached. But the solution 
which was cooled was a saturated solution, so that the ice and salt 
must have separated out in the proportions in which they existed in 
this saturated solution, i.e. the solution solidifies like a single sub- 

The same result is arrived at if we follow the curve for the cooling 
of a strong solution of salt instead of a weak one. When the solution 
is cooled, a point is at last reached at which the solution is 
saturated. The result of further cooling is that some of the salt 
separates out, the solution being now saturated for the lower tempera- 
ture. As we cool the solution, therefore, it becomes less and less 
concentrated, the concentrations following the solubility or saturation 
curve on the right-hand side of the diagram. At last a point is 
reached where the saturation curve cuts the freezing-point curve. If 
we could follow the solubility curve below this point, further cooling 
would cause more salt to separate, but the weaker solution resulting 
would now be below its freezing point, and consequently ice would 
begin to separate out, and this would continue until the concentration 
of the solution was restored to the original value it possessed where 
the two curves intersect. If we take a solution of the particular con- 
centration given by the point of intersection of the freezing point and 
saturation curves, we may cool it without the separation of either 
salt or ice until we come to the temperature corresponding to the 
intersection. If we continue to abstract heat, the solution solidifies 
as a whole, both ice and salt separating in the proportions in which 
they exist in the solution, without any change of temperature taking 
place until all is solid. The solution then behaves like a pure liquid, 
and has a definite freezing point. For this reason the separated 
substance was supposed to be a definite compound of salt and water, 
and was called a cryohydrate. It is now known, however, that the 
salt and ice separate out independently of each other and not in the 
form of a compound. Cryohydric solutions, in fact, are quite analogous 
to the constant-boiling mixtures referred to in Chapter IX. ; they are 
constant-freezing mixtures. 

It can be easily seen from a study of the diagram that no solution 
of salt in water can exist in a stable state below the cryohydric tem- 
perature, and that this temperature is the lowest that can be produced 
by mixing ice and salt, or snow and salt together. If we consider a hori- 
zontal line cutting the diagram at the temperature 0°, we see that there 
is only a certain range of concentrations, viz. from per cent to 26*3 per 
cent possible for salt solutions, for above the latter concentration we 
should have supersaturated solutions, and salt would separate. A similar 
line at - 1 0° cuts both curves, and it is only the part between the two 
curves, viz. from 14 per cent to 25 per cent, that represents possible 



concentrations for salt solutions. If we had a stronger solution than 
25 per cent, it would be supersaturated at that temperature, and salt 
would fall out ; if we had a weaker solution at that temperature than 
14 per cent, it would be below its freezing point, and ice would separate. 
Taking horizontal lines at still lower temperatures, the parts enclosed 
between the two curves, Le. the range of possible concentrations, 
become smaller and smaller, until at the cryohydric temperature 
it has shrunk to a point. At that temperature, therefore, there is 
only one salt solution possible, and below that temperature none is 
possible at all, unless indeed it were an unstable supersaturated 
or superfused solution, which when brought into contact with the 
corresponding solid would immediately assume the temperature and 
concentration of the cryohydric solution. 

At and below the cryohydric temperature, ice and salt can exist 
together without affecting each other ; and at the cryohydric tempera- 
ture they can also exist together with the cryohydric solution. 
Above the cryohydric point, ice can only exist in contact with a 
solution having one definite concentration, and salt with a solution 
having another definite concentration. If we mix therefore ice, salt, 
and water at a temperature above the cryohydric point, and keep 
them well mixed, there can be no real equilibrium. The water will 
tend to dissolve salt until it becomes saturated, but this saturated 
solution will not be in equilibrium with ice, being too concentrated 
(compare the diagram at -10° say). Ice will therefore melt and the 
solution will become more dilute, and again dissolve more salt, with 
the result that more ice will melt. Now a certain amount of heat 
must be supplied in order to melt ice. If this heat is not supplied 
from an external source it must be derived from the mixture itself. 
The temperature of the mixture will therefore fall, and if we continue 
to mix, the same causes will effect a further lowering of temperature 
until the cryohydric point is reached, at which one and the same 
solution can be in equilibrium with both ice and salt. The solution 
adjusts itself to this concentration, and ice and salt now melt (if heat 
from external sources reaches the mixture) in the proportions of the 
cryohydric solution, and the temperature will remain constant at the 
cryohydric point until all the ice or all the salt has disappeared. It 
may happen, of course, that the original proportions of ice, salt, and 
water may have been such that one of the solids will disappear before 
the cryohydric point is reached. No further cooling can occur after this 
disappearance takes place, and the mixture is not then a satisfactory 
freezing mixture. In order that a freezing mixture may be effectual 
and reach the cryohydric point, thorough mixing of the ingredients 
is also necessary, for on this depends the attainment of the final 
equilibrium. For this reason snow and salt form a better freezing 
mixture than pounded ice and salt, for in the latter case the mixing can 
scarcely be so thorough. 


When ice and salt are brought into contact at a constant 
temperature below the freezing point of water but above the cryo- 
hydric point, they form a liquid solution at the points at which they 
come in contact, and this solution strives to adjust itself so as to be in 
equilibrium with both solids, with the result that one of the solids 
entirely disappears. If, as is the case in salting snow or ice-covered 
streets, the solution assumes the general ground temperature, both 
solids disappear when enough salt has been added. The quantity 
of salt necessary to melt a given quantity of ice depends on the 
temperature of the ice, which is practically the temperature of 
the ground immediately below it. This quantity can easily be cal- 
culated from the above diagram. Let the temperature of the ice 
be - 2° C. From the diagram we find that the solution which freezes 
at this temperature contains about 3 per cent of salt. If we add less 
salt than amounts to 3 per cent of the weight of ice to be removed, the 
whole of the ice will not be melted, but only enough for the formation of 
a solution of the above strength, this solution then being in equilibrium 
with the remaining ice. If we add 3 per cent of the weight of the ice, all 
the latter will melt. If we add more then 3 per cent of salt, the ice will 
all melt and the solution will dissolve the excess of salt, unless the excess 
is so great that the solution becomes saturated before the whole excess 
is dissolved. If the temperature of the ground (and the ice) is below 
the cryohydric point, no amount of salt, however great, will melt the 
i ice. It should be noted that the circumstances of the case are here 
somewhat different from those under which a freezing mixture is 
usually formed. In the latter case conduction is avoided as far as 
possible, as the object is to keep the temperature down. If, on the 
other hand, excess of salt is thrown on the streets, although the 
temperature may for a short time sink below .the temperature of the 
street, conduction from below speedily restores it to its original value. 

In the above discussion it will be observed that there is no 
practical difference between the behaviour of the solutions towards the 
ice and towards the salt. The solution which can remain unchanged 
in contact with ice may be said to be saturated with regard to ice, just 
as the solution which can remain unchanged in contact with common 
salt is said to be saturated with regard to the salt. This serves to 
illustrate the conventional nature of the distinction between solvent 
and dissolved body. As soon as we reach a temperature at which 
both substances are solid, the solution bears for all practical purposes 
the same relation to both. 

What has been said with regard to salt and water holds equally 
well for other substances, where we have two concentration curves 
meeting. The hydrates of many inorganic salts afford excellent 
instances in point. As an example we may take the hydrates of 
•ferric chloride. In the diagram temperatures are tabulated as before 
on the vertical axis and concentrations on the horizontal axis. 




The concentrations, however, in this case are expressed not in 
parts of salt dissolved in 100 parts of water, but as the number of 

molecules of ferric chlor- 
ide (Fe 2 CI 6 ) to 100 mole- 
cules of water. Ferric 
chloride is usually met 
with as the yellow 
hydrate Fe 2 Cl 6 . 12 aq. 
or as the black metallic- 
looking scales of the 
anhydrous salt. It also 
exists, however, in the 
form of hydrates, having 
the formulae Fe 2 Cl 6 .7aq., 

Fe 2 Cl fl 
Fe 2 Cl 6 . 

6 10 15 20 25 30 

Mols. Fe 2 Cl 6 to 100 Mots. H 2 

Fig. 10. 

aq., and 
4 aq. The dia- 
gram shows the solu- 
bility of all these 
hydrates. Each hydrate 
has its own solubility 
curve, and these curves 
cut each other at various 
points. Towards the left 
35 of the diagram is given 
the freezing-point curve 
of dilute ferric chloride 
solutions, water being 
considered the solvent. The curve cutting this is the solubility curve 
of the hydrate Fe 2 Cl 6 . 1 2 aq. On the left of this curve we have pre- 
cisely the kind of diagram already given on p. 64, with the difference 
that the solid salt is now a hydrate of ferric chloride instead of 
anhydrous sodium chloride. The point where the curves cut is the 
cryohydric point, and gives the lowest temperature that can be got' 
by mixing ice and FegCl^ . 1 2 aq. To the right of the cryohydric 
point it will be observed that the curve of the dodecahydrate cul- 
minates at a point A, declining from this maximum as the con- 
centration increases. If we draw a horizontal line at a temperature 
a little below 40°, it will cut the curve of the dodecahydrate in 
two places, i.e. at one and' the same temperature this hydrate has 
apparently two solubilities, or can be in equilibrium with two ferric 
chloride solutions of different concentrations. This is not an un- 
common occurrence with hydrates, and may be most easily under- 
stood by a reference to the maximum point. The concentration here 
is the same as the composition of the solid hydrate i.e. 1 molecule 
Fe 2 Cl 6 to 12 molecules of water. The solution then may be looked 
upon as the melted hydrate, and the maximum temperature as 


the melting point of the hydrate. But, as we have seen, ice is in 
equilibrium in contact with water plus a foreign substance at a lower 
temperature than when in contact with water (i.e. melted ice) alone. 
The melting point of ice therefore is highest when it is in contact with 
a liquid of its own composition : it melts at a lower temperature when 
the liquid with which it is in contact contains a foreign substance. 
This holds true generally — a substance always melts at the highest 
temperature when in contact with the liquid produced by its own 
complete fusion. Now in the above case the maximum melting point 
of the hydrate Fe 2 Cl 6 . 1 2 aq. occurs when the concentration of the 
solution corresponds to this formula. If the solution contains either 
ferric chloride or water in excess of this ratio, the dodecahydrate will 
melt at a lower temperature. The curve therefore falls both to the 
right and the left of the concentration given by the ratio of Fe 2 Cl 6 to 
water in the dodecahydrate, with the result that there are two 
different solutions with which the hydrate can be in equilibrium at 
one and the same temperature. One of these solutions contains more 
water than the solid with which it is in equilibrium — the other 
contains more ferric chloride. The branch of the curve to the left of 
the maximum may evidently either be looked on as a solubility curve 
of the dodecahydrate in water, or as a melting-point curve of the 
dodecahydrate in contact with solutions of varying concentration. 
The branch to the right we usually regard as a melting-point curve ; 
but it also may be looked on as a solubility curve. 

If we follow now the right descending branch of the curve of the 
dodecahydrate, we find that we reach a point where it cuts the curve 
of another hydrate, the heptahydrate Fe 2 Cl 6 . 7 aq. From inspection 
of the diagram, the point of intersection is evidently of the same 
nature as the cryohydric point, where the curve for the melting point 
of ice intersects the curve for the melting point of the dodecahydrate. 
At the cryohydric point proper, ice and the solid dodecahydrate 
can exist together with each other and with a solution of a certain 
concentration : at the other point of intersection, the dodecahydrate 
and the heptahydrate can exist together and in contact with another 
solution of definite concentration. If a solution of this particular con- 
centration is cooled, it solidifies as a whole, and the freezing point remains 
constant. As before, however, the solid which separates out is not a 
pure substance, but a mixture of solids — the two hydrates. Proceeding 
farther to the right, we again reach a maximum and again dip to a 
"cryohydric" point, viz. the point where the curves of the heptahydrate 
and the pentahydrate intersect. In still more concentrated solutions 
the same phenomena are repeated, until we finally come to a cryo- 
hydric point where the curve of the tetrahydrate cuts the solubility 
curve of the anhydrous salt, after which we have a solubility curve of 
the ordinary type. Every maximum corresponds in composition to a 
definite hydrate, and in temperature to the melting point of that 



hydrate, the solid melting at a definite and constant temperature. 
At every intersection we again have a composition corresponding 
to a constant melting point and freezing point ; here, however, it is 
not a definite solid that melts or separates out, but a mixture of two 

It should be noted that whenever we have a constant freezing 
point, the composition of the liquid and of the solid is the same. If 
the composition of the solid which separates is not the same as that of 
the liquid from which it separates, the temperature will fall as the 
freezing goes on. 

It is often possible to follow the curves of hydrates below their 
points of intersection, as the dotted continuations in the figure indicate. 












., 1 ,,_.! 

20 40 60 

Percentage Hsxachlor 
Fig. 11. 







The phenomena here are of the same nature as those already referred 
to on page 65. The solution which is saturated with respect to the 
dotted hydrate is unstable, or rather "metastable" (cp. Chap. XL), 
and may suddenly deposit another hydrate. 

It occasionally happens that when two substances are melted 
together, the liquid when cooled does not deposit one of the substances 
only, but both at once. This we have seen to be the case with 
salt solutions having the cryohydric composition, but it is only when 
the liquid has this peculiar concentration that the deposition of both 
solids occurs. With other substances, however, both solids may be 
deposited simultaneously, no matter what the composition of the liquid 
is. If the substances are always deposited in the proportions in which 
they exist in the liquid, the liquid will have a constant freezing point. 
An instance of this kind apparently exists in the case of two substances 




recently investigated. These are closely related chemically, one 
being the hexachlor-derivative of a cyclopentenone, and the other the 
corresponding pentachlor-momobrom- derivative. It is only in cases 
when the substances are thus closely related that we may expect 
behaviour of this kind. The diagram Fig. 11 gives temperatures 
(melting points) on the vertical axis and compositions (molecular 
percentages) on the horizontal axis. The curve is practically a straight 
line joining the melting points of the two substances. No matter 
what the composition of the liquid may be, it freezes as a whole and 
has a constant melting point. 

If the solid mixture which separates out has not the same 
composition as the liquid, but a different one, which varies with the 

varying composition of the residual liquid, the temperature will fall 
as solidification progresses. Should a point be reached where the com- 
position of the solid separating is the same as that of the residual 
liquid, the liquid will then freeze as a whole. 

That mixture which melts at a lower temperature than any other 
mixture of the same substances is called the eutectic mixture. 
The accompanying figure gives a curve of melting points of mixtures 
of stearic and palmitic acids (Fig. 12). Stearic acid melts at 69*0°, 
palmitic acid at 62*0°, and there is an eutectic mixture, containing 
about 70 per cent of palmitic acid, which melts at 55*0°. Here the 
two curves do not cut each other at an angle, as is the case at cryo- 
hydric points ; we have apparently rather a continuous curve which 
dips to a minimum and then ascends. This case corresponds to that 
referred to above, in which neither of the substances separates out 
in the pure state. 


In the ordinary method of melting-point determination practised 
by organic chemists, it is well known that pure substances give sharp 
melting points, whilst impure substances begin to soften and melt at 
a temperature several degrees lower than the temperature at which 
they are completely liquefied. This corresponds with the general rule 
that the presence of a foreign substance lowers the freezing point, the 
melting point being similarly lowered. In many cases when solids 
are brought into contact with each other the melting point of one of 
them will be depressed. Thus it is possible to liquefy a mixture of filings 
of cadmium, zinc, lead, and bismuth by merely leaving them in contact 
at 100°, although all of them have melting points much above 100°. 
If a solid, therefore, is mixed with another solid as an impurity, its 
melting point will very likely be lowered, especially when, as usually 
happens, the impurity is a chemically similar substance. In this case 
the two will probably be mutually soluble when liquid, and mutual 
solubility is a necessary condition for lowering of the melting point. 
If the substances are not mutually soluble, e.g. sand and sulphur, the 
melting point will not be affected. 



We have seen in Chapter IV. that when a gas is subjected to great 
pressures it occupies a small volume compared to that which it 
occupies at the ordinary pressure, although a larger volume than it 
would occupy if it contracted exactly according to Boyle's Law 
(cp. p. 27). Every gas, above a certain temperature which is 
characteristic for it, exhibits this behaviour, but if the gas be below 
the characteristic temperature, constantly increased pressure will at 
length convert it into a liquid. The temperature below which the 
substance can be condensed into a liquid, and above which it under- 
goes compression without liquefaction, is called the critical tempera- 
ture of the substance, and the pressure which at the critical temperature 
just suffices to condense the gas to the liquid form, is called the critical 
pressure. Under these conditions the substance has a certain 
definite critical density, the reciprocal of which is the critical 
volume. The following table gives the critical temperature and 
pressure of various substances, the temperatures being in degrees 
centigrade and the pressures in atmospheres : — 

Critical Temperature. 

Critical Pressure. 







Argon . 



Oxygen . 



Methane . 

- 81'8 


Carbon dioxide 

+ 31*35 


Nitrons oxide . 



Sulphur dioxide 



Ethyl ether . 



Ethyl alcohol . 



It will be observed that the critical pressure, which is the pressure 
necessary to liquefy the gas at the highest temperature at which the 


liquefaction of the gas by pressure is possible, in no case in the above 
table exceeds 100 atmospheres, being in general very much less than 
this. At lower temperatures than the critical temperature, a smaller 
pressure than the critical pressure effects the liquefaction. We may 
indeed say, as a general rule, that if a gas under any given condi- 
tions is not liquefied by a pressure of 100 atmospheres, it will not 
under these conditions be liquefied by any pressure, however great. 
Needlessly high pressures were employed when the liquefactions of 
the so-called permanent gases (oxygen, nitrogen, and the like) were first 
successfully attempted, 500 atmospheres being not uncommon. 

For the liquefaction of a gas a low temperature is the necessary 
condition, not a great pressure. If the gas is not cooled to its 
critical point, no pressure which can be applied is capable of liquefying 
it, while at temperatures not far below this point the gas may be 
liquefied at the atmospheric pressure. All known gases have now been 
obtained in the liquid state, with the doubtful exception of helium, 
and the liquefaction in all cases can be effected by cooling alone. 

One of the methods by which oxygen and similar gases were first 
successfully liquefied was to cool the gas to as low a temperature as 
possible while it was subjected to great pressure, and then suddenly to 
release the pressure. The gas expanded, and in doing so performed 
work. An amount of heat equivalent to this work must therefore 
have been supplied. This heat was partially abstracted from the gas 
itself, with the result that a portion of it liquefied owing to the fall of 

A method which has of late been worked with conspicuous success 
to liquefy the least condensible gases also depends on the expansion 
of the gases by diminution of pressure, but the principle involved is 
entirely different from that just referred to. When any actually exist- 
ing gas is made to pass, at a suitable temperature, through a porous 
plug or valve from a high pressure to a lower pressure, its temperature 
is slightly diminished; and if the process is repeated, the gas gets 
colder and colder, until finally its point of liquefaction is reached. 
Now while this is true for existing gases, it is not true for an ideally 
perfect gas. For such a gas there would be no cooling (Joule-Thomson 
effect), and liquefaction in this way would be impossible. The cooling 
here observed is due to deviations from the simple gas laws, and must 
therefore never be confused with the cooling produced by the expansion 
of a gas under suddenly released pressure, which depends chiefly on the 
external work done, and holds good for perfect and imperfect gases 
alike. A diagram illustrating the principle of Linde's apparatus is given 
in Fig. 13. The inlet for the gas is on the left of the diagram at I. 
The gas passes through the pump from the pressure P (say 40 atm.) to 
the pressure P' (say 200 atm.), and, following the direction of the arrows, 
passes down the central tube C to the throttle valve T, where the pres- 
sure falls again to the original pressure P. In passing through this valve 




the gas becomes colder, and circulating backwards by means of the side 
tube and the annular space A, it cools the next portion of gas coming 




u t 




— P 

downwards through C 

to the throttle valve. 

This next portion of 

gas, after it has passed 

through the valve, is 

cooled to a lower tem- 
perature than the first 

portion, and serves in 

its turn to cool the 

fresh gas arriving by 

C. The temperature 

in the neighbourhood 

of the throttle valve 

thus constantly sinks, 

and finally the point 

of liquefaction of the 

gas is reached. Liquid 1-^30: 

appears at the nozzle 

N and collects at L, 

the place of the lique- 
fied gas being supplied 

by fresh quantities ad- FlG 13 

mitted through I. 

All liquids tend to assume the gaseous state, and the measure of 

this tendency is what we call the vapour tension of the liquid. The 
tendency is exhibited in very different degree by different liquids. 
Water if left in an open vessel evaporates slowly ; alcohol under the 
same conditions evaporates much more rapidly, and ether more 
rapidly still. Mercury, on the other hand, scarcely evaporates at all 
at the ordinary temperature. That it does so slowly, however, may 
be proved by suspending a piece of gold leaf a little distance above a 
mercury surface. After some time the gold leaf will be found to be 
amalgamated, indicating that the mercury has reached the gold in a 
state of vapour and combined with it to form a gold amalgam. 

For a given liquid there corresponds to each temperature a certain 
definite pressure of its vapour, at which the two will remain in contact 
unchanged. At that pressure and temperature none of the liquid will 
pass into vapour ; nor will any of the vapour condense to form liquid. 
The gas pressure of the vapour balances the vapour tension of the 
liquid, and this gas pressure is said to be the vapour pressure of the 
liquid at that temperature, and the vapour itself is said to be saturated, 
For all liquids the vapour pressure increases with rise of temperature, 
and consequently all liquids evaporate more readily as the temperature 
is raised. When the vapour tension of the liquid just exceeds the 


external pressure (usually the atmospheric pressure) on the liquid 
surface, the liquid passes freely into vapour, and is said to boil. 
If we heat a liquid in a closed space of appropriate dimensions, 
the pressure within the space rises, owing to the increase of the 
vapour pressure of the liquid, and the properties of the liquid 
and its saturated vapour gradually approximate to each other, until 
at last, when the critical temperature is reached, the liquid and 
the vapour become identical, and all distinction between them dis- 
appears. 1 The pressure registered is then the critical pressure of 
the substance. 

Having considered above the effect of raising the temperature of a 
substance at constant volume, we may now pass to the consideration 
of the effect of varying pressure at constant temperature. Let us 
imagine the vapour of a liquid enclosed in a cylinder with a movable 
piston, and let the pressure on the piston be less than the vapour pres- 
sure of the liquid at the constant temperature considered. If the 
pressure on the piston is gradually increased, the gas will be compressed 
into smaller bulk at a rate greater than would accord with Boyle's Law, 
till the moment at which the vapour pressure is reached, when the liquid 
will begin to make its appearance. If the piston is still pressed home 
the gas will now pass into the liquid state without any further increase 
of pressure being necessary. The pressure necessary to liquefy a gas 
at any given temperature needs therefore to be only slightly greater 
than the vapour pressure of the liquid at that temperature, which 
must be always less than the critical pressure, for this is the upper limit 
of the series of vapour pressures (cp. p. 99). 

If on two axes at right angles to each other we plot the corre- 
sponding pressures and volumes of a quantity of gas which obeys 
Boyle's Law, we should get a rectangular hyperbola as isothermal 
curve, or curve of constant temperature. Now all gases diverge more 
or less from this law, and the isothermal curves obtained only 
approximate to rectangular hyperbolas when the gas is not near the 
critical condition. A consideration of Andrews' jpv diagram for 
carbonic acid will be instructive (Fig. 14). The critical temperature of 
carbon dioxide is about 31*1°. At a temperature of 48° the pv curve 
runs without any flexure, and though it is not a rectangular hyperbola 
the volume diminishes regularly with increase of pressure. At tem- 
peratures nearer the critical pressure the curves show less regularity, 
exhibiting at a certain pressure (about 80 atmospheres) contrary flexure, 
although at lower and higher pressures they are quite regular. At the 
critical temperature the curve runs for a very short distance parallel 

1 The student may compare this behaviour with what happens when aniline and 
water are heated together in a closed space (p. 53). At first there are two layers, but 
as the temperature is raised, these layers approximate to each other in composition (Fig. 
5), density, etc., until at 165° they become identical, the whole being then homogeneous. 
The lowest temperature of complete miscibility has been called the "critical solution 
temperature " for the pair of liquids. 




to the v axis, when the pressure is 73 atmospheres. At a still lower 
temperature, when the pressure is increased slowly, the volume di- 
minishes somewhat rapidly, 

at a 


the curve 

certain pressure 

suddenly breaks 

runs horizontally, so 

no further rise of 

Fig. 14. 

pressure is required to effect 
a diminution of volume. 
This takes place when the 
liquid and gas coexist, and 
the pressure is then the 
vapour pressure of the 
liquid. When that pressure 
is reached, the liquid appears 
and goes on increasing in 
quantity by the liquefaction 
of the gas without further 
rise of pressure until the gas 
has entirely disappeared, 
when the curve suddenly 
bends upwards from the 
horizontal. In the liquid 
a very small change of 
volume now follows from a 
considerable rise of pressure. 

At a lower temperature still, the same phenomena are observable : 
the formation of liquid taking place at lower pressures, for, as we 
have already seen, the vapour pressures of all liquids diminish with fall 
of temperature. In each curve there are two breaks, first when the 
curve becomes horizontal, and second when it ceases to be horizontal. 
The value of the pressure at the horizontal portion is the vapour 
pressure at the particular temperature for which the curve is 
constructed. If all the points where the breaks occur are connected, 
we get a " border curve " which is shown by the dotted line in the 
figure. The region within this border curve gives the pressures and 
volumes at which the gas and liquid can coexist. On the right of 
the border curve and above it the substance is a gas : on the left it is 
a liquid. If we so change the temperature, pressure, and volume, 
that the corresponding points in the diagram keep entirely outside the 
border curve, liquid and gas never exist together, and there is no 
discontinuity in the passage from the liquid to the gaseous state, or 
vice versa. Let us, for example, take liquid carbon dioxide at the 
temperature 20°, and at the pressure and volume indicated by the 
point A. Let us raise the pressure of this liquid at constant tem- 
perature until it passes the critical pressure, and then, keeping the 


pressure constant, let us increase the temperature until it is greater 
than the critical temperature, say 48°. At no point during the rise of 
temperature does the liquid change its state suddenly ; it passes with 
perfect continuity into the state of gas. By expanding now at constant 
temperature along the isothermal of 48°, and then cooling at constant 
volume, we finally come to the .same temperature curve as that on 
which we started, viz. 20°, and so have converted the liquid into gas 
at the same temperature without sudden change of state. 
- This gradual passage from the gaseous to the liquid state, and vice 
versa, shows us that these states are not so sharply defined from each 
other as we are in general disposed to imagine, but are rather united 
by a continuous series of intermediate states. The sharp definition 
that we are accustomed to perceive is a consequence of the ordinary 
conditions of temperature and pressure at which we work, rather than 
of any inherent properties of the substance operated on. In Chapter 
X. reference will be made to the theory of van der Waals, which deals 
with the continuity of the liquid and gaseous states in a systematic 

Solids have often, like liquids, a measurable vapour pressure. On a 
windy day, snow and ice may be seen to disappear by evaporation, 
although the temperature may be far below the freezing point. The 
solid here passes into vapour directly without first assuming the liquid 
state, and the vapour is removed by the wind as it is formed. A piece 
of camphor when left in the open air gradually diminishes in bulk and 
finally disappears, owing to evaporation without melting. Since the 
vapour pressure of the solid is always less than the vapour pressure 
of the liquid, the vaporisation in the former case takes place more 

Solids are said to sublime when on heating they pass directly 
into vapour, which on being cooled does not condense to a liquid but 
directly to a solid. Sublimation takes place with ease under ordinary 
conditions when the solid has at its melting point a vapour pressure 
not far removed from the external pressure, or, what practically comes 
to the same thing, when the melting and boiling points of the 
substance are comparatively close together. Arsenic trioxide, when 
heated gradually at the ordinary pressure, passes directly into vapour 
without assuming the liquid state, and can only be made to melt if 
heated rapidly under increased external pressure. On cooling, the 
vapours pass at once into the solid form. By sufficiently reducing the 
external pressure, the boiling point of any substance can always be 
lowered to the neighbourhood of its melting point, and the sublima- 
tion can take place. Ice, for instance, can be easily but slowly sub- 
limed at a temperature below the freezing point from one part of an 
exhausted vessel to another, the ice passing directly into water vapour 
and the water vapour into ice. In the formation of snow the water 
vapour assumes the crystalline state directly. Sublimation can in any 


case be made to occur if the substance is brought to a temperature 
just below the melting point of the solid, and the vapours derived from 
it are rapidly cooled. 

It has been observed that substances volatilise to a greater extent 
in a gas than they do in a perfectly exhausted space. It is usually 
assumed for rough purposes that gases exert no influence on each other, 
the sum of the values of a physical property for two separate gases 
remaining the same when the gases are mixed. This cannot be 
accurately the case, since we are obliged to conclude that the molecules 
even of the same gas influence each other (see Chap. X.) The 
reciprocal influence of the gas molecules will evidently be greatest 
when the molecules are closely packed together, i.e. at great pressures, 
but it may be noticeable even at ordinary pressures with sufficiently 
refined means of measurement. 1 If we consider that a substance like 
water must at temperatures just below the critical temperature exert 
a solvent action on many other substances as long as it remains a 
liquid, and if we further consider that saturated water vapour at such 
temperature does not sensibly differ in its properties from liquid water, 
seeing that the properties of the saturated vapour and the liquid become 
identical at the critical point itself, we shall not find it surprising that 
water vapour under these conditions exerts a solvent action even on 
solids. It is a well -ascertained fact that gases under high pressure 
exert a solvent action on solids, and the solvent action of gases under 
ordinary pressure can only differ from this in degree. It is quite in 
accordance with our theoretical knowledge, then, that a substance such 
as iodine should be more volatile in air than in a very perfect vacuum, 
as Dewar has recently ascertained. The air here acts as " solvent," 
and indeed we can always look on a mixture of two gases as a solution 
of one in the other, although the influence of the " solvent n is not 
marked as it is in liquid solutions. 

When a substance is dissolved in a liquid, the vapour pressure of 
the latter is lowered at all temperatures, and the lowering for small 
amounts is approximately proportional to the quantity of substance dis- 
solved in a given amount of the liquid (Wiillner's Law). If we take, 
for example, a fairly dilute aqueous solution of a very soluble substance 
such as sugar or calcium chloride (which are not themselves volatile), and 
evaporate it on a water bath at 100°, the evaporation will at first proceed 
rapidly, for the vapour pressure of the dilute solution is not much less 
than that of pure water, being therefore at the beginning nearly equal 
to one atmosphere. As the evaporation proceeds, however, the solution 

1 According to Dalton's law of partial pressures, the total pressure of a mixture of 
gases is equal to the sum of the pressures which the different gases would exert if each 
separately occupied the whole space afforded to the mixture. Even for moderate pres- 
sures this law is not absolutely exact. A better approximation to the experimental data 
is given by the following rule : — If we measure the volumes of the separate gases and of 
the mixture at the same pressure, the sum of the volumes of the component gases is equal 
to the volume of the mixture. 


becomes more concentrated, and the vapour pressure for 100° becomes 
continually less. The evaporation therefore progresses more slowly, 
and towards the end the vapour pressure of the water in the solution 
is so low that practically no vapour comes off at all and the solution 
cannot be further concentrated. If the solid, on the other hand, is not 
very soluble and begins to separate out on evaporation, the solution 
soon reaches its maximum concentration, and the vapour pressure no 
longer diminishes, so that the evaporation proceeds at a uniform rate 
until all the water has been driven off. To get a given vapour 
pressure of solvent from a solution, the solution must be heated to a 
higher temperature than would be required for the pure solvent, since 
at a given temperature the vapour pressure of the solution is lower 
than that of the solvent. A consequence of the diminution of the 
vapour pressure of a liquid by the dissolution in it of some foreign 
substance is therefore that the boiling point of the solution is 
higher than that of the pure solvent. We may thus obtain an 
aqueous liquid of higher boiling point than 100° by dissolving some 
fairly soluble non-volatile substance in water. By employing common 
salt, we obtain a brine bath, which is sometimes used instead of a 
water bath for heating substances to temperatures slightly above 100°. 
It should be noted that in this case the vessel to be heated must be 
immersed in the brine, for the steam from the boiling solution will only 
heat it to 100°, although the temperature of the salt solution may be as 
high as 1 1 0°. As the boiling point of a saturated solution of calcium 
chloride is 180°, this substance may be used when still higher tempera- 
tures are required. 

Distillation of Mixtures. — We supposed in the preceding para- 
graph that the dissolved substance was non-volatile. When the dis- 
solved body is volatile as well as the solvent, Bach substance lowers 
the vapour pressure of the other, and the boiling point of the mixture 
may be higher or lower than the boiling point of either, depending on 
the sum of the vapour-pressure values of the two substances, tf, for 
example, we take a mixture of alcohol and water, the vapour given off 
consists of a mixture of the vapours of these liquids, and the mixture 
boils when the temperature is such that the sum of the two partial 
vapour pressures is equal to the atmospheric pressure. In general, the 
composition of the remaining mixture will not be the same as the com- 
position of the vapour evolved, so that as the evaporation or distillation 
goes on, the temperature of ebullition changes, since for each mixture 
a different temperature is required in order that the sum of the vapour 
pressures may reach the constant atmospheric pressure. It is evident 
that in the process of distillation as ordinarily conducted the boiling 
point of the mixture must gradually rise, since the more volatile portions 
evaporate first. If the original mixture contains much water and little 
alcohol, the boiling point will gradually rise as the more volatile 
alcohol (boiling point 78°) distils off, and finally pure water will 


remain behind when the boiling point reaches 100°. We can therefore, 
as a rule, free an aqueous liquid from alcohol by heating on a steam 
bath. If, on the other hand, the solution contains much alcohol and 
little water, it is practically impossible to get rid of all the water by \ 
distillation. At the boiling point of the alcohol, pure water has no I 
doubt a considerable vapour pressure, but when the mixture consists 
nearly wholly of alcohol, the vapour pressure is lowered very much, 
and it is increasingly difficult to get perfect separation of the two 
constituents as the quantity of water in the liquid gets less and less. 
By properly constructed fractionating apparatus it is, however, easy 
to get a distillate containing less than 5 per cent of water, and theoreti- 
cally a perfect separation is possible. 

The mixtures of some" liquids differ from the example given above 
inasmuch as they tend to separate on distillation, not into the two 
components, but into one of the components and a constant boiling 
mixture of both. This constant boiling mixture can be distilled 
unchanged, the composition of the distillate being the same as the 
composition of the residual mixture. Examples in point may be found 
in the aqueous solutions of propyl alcohol and of formic acid. The 
constant boiling mixture is that which has either a greater vapour 
pressure than that of any other mixture, or a less vapour pressure 
than that of any other mixture. Thus a mixture of propyl alcohol 
and water containing 70 per cent of the former has, under ordinary 
conditions, a greater vapour pressure than any other mixture of the 
two substances. It is therefore the mixture of minimum boiling point 
at atmospheric pressure, and if we distil any mixture whatever of 
the two substances, the distillate will always approximate more 
closely in composition to this mixture than will the residue in the 
distilling flask. On the other hand, a mixture of formic acid and water 
containing 75 per cent of the acid has a lower vapour pressure than 
any other mixture, i.e. it is the mixture with the highest boiling point. 
If, then, we distil any given mixture of formic acid and water, the 
composition of the residue will always approximate more closely to 
this mixture of maximum boiling point than will the composition of 
the distillate. 

Certain constant boiling mixtures (mixtures of maximum boiling 
point) were for long looked upon as true chemical compounds, for 
their composition was not affected by distillation. Nitric acid, for ' 
example (boiling point 86°), forms with water a constant boiling j 
mixture which contains 68 per cent of acid and boils at 126°. If a 
weaker acid than this is distilled, water comes over in excess, and the \ 

I residue eventually attains the above composition and boiling point. \ 
If a stronger acid than this is distilled, the distillate contains excess of \ 
nitric acid, whilst the residue grows weaker and the boiling point rises 

\ until the values for the constant boiling mixture are attained. The dilu- 
tion of the residue is assisted in the case of strong nitric acid by the 



partial decomposition of the acid into oxygen, nitrogen peroxide, and 
Avater, the water remaining for the most part in the distilling apparatus. 
Hydrochloric acid forms with water a similar mixture of constant 
boiling point. This mixture contains 20*2 per cent of acid, and distils 
unchanged at 110° under atmospheric pressure. That such constant 
boiling liquids are mixtures and not chemical compounds is proved by 
the composition of the liquid which distils unchanged at any one 
pressure varying with the pressure at which the distillation is conducted. 
Thus the mixture which distils unchanged at 2 atm. contains 19*0 per 
cent of hydrochloric acid, differing therefore in composition from the 
constant boiling mixture under ordinary pressure. The same error of 
assigning definite chemical union to mixtures which pass unchanged 
through some physical process has often been committed, and instances 
of this nature have already been referred to (cp. p. 65). 

When liquids that are only partially miscible are distilled 
together, a distillate of definite composition is obtained as long as two 
separate layers are present, for, as can be easily shown, the same 
vapour is given off by each of the two layers, and the only result of 
distillation is to change the relative quantities of the two layers, the 
boiling point remaining constant until one of the layers vanishes, after 
which the distillation proceeds as in the case of two completely 
miscible liquids. 

When two liquids that are completely immiscible are subjected 
to distillation from the same vessel, neither influences the vapour 
pressure of the other ; and when the sum of their vapour pressures is 
equal to the external pressure, distillation goes on without change in 
the composition of the distillate until one of the liquids disappears. 
Nitrobenzene and water may serve as an instance of a pair of nearly 
immiscible liquids. The mixture boils at 99° under a pressure of 
760 mm. Now water at this temperature has a vapour pressure of 
733 mm. ; the remaining 27 mm. is therefore due to the vapour 
pressure of the nitrobenzene. Although the pressure exerted by the 
nitrobenzene is thus relatively small, the weight of nitrobenzene which 
distils over with the water is considerable ; and it is this circumstance 
which renders distillation with Steam, an operation so often em- 
ployed in organic chemistry, a practical success. The weights may be 
calculated from the vapour pressure by means of Avogadro's Law. 
The molecular weight of water is 18 ; the molecular weight of nitro- 
benzene is 121. If we consider the weight of the gram-molecular 

2^*4 ^273 + 99^ *""" *** 

volume at 99°, viz. — — ^^ litres, the mixed vapour will be 

, . , 18x733 . . , 121 x27 

found to consist of — =^r — g. of water vapour and — ^FR — & 

of nitrobenzene vapour. The ratio of the weight of water to nitro- 
benzene in the vapour is then 18x733 to 121x2 7, or roughly 
4 to 1 ; and this is the ratio of the weights in the distillate. Thus, 


although nitrobenzene has only 1/27 of the vapour pressure of water 
at the boiling point of the mixture, one-fifth of the liquid collected is 
nitrobenzene. This, of course, is due to the molecular weight of the 
water being so much smaller than that of the nitrobenzene. If an 
organic substance is unaffected by water and has a vapour pressure of 
even 10 mm. at 100°, distillation with steam for purposes of purifica- 
tion will in general be repaid. For although its vapour pressure may 
be only an insignificant fraction of that of water, the higher molecular 
weight makes up for this, and appreciable quantities come over and 
condense with the steam. It is thus the low molecular weight of 
water which renders it so specially suited for vapour distillation. 



In the preceding chapters we have seen that a consideration of the 
composition and properties of substances, and the changes which they 
undergo, has led to the conception of atoms and molecules ; but as yet 
we have not dealt with the mechanical constitution of these substances, 
in other words, we have not considered how the molecules go to build 
up the whole — whether they are at rest or in motion, or whether in 
the different states of matter there is a difference in the state of the 

It is plain that the kind of matter most suitable for study from 
this point of view is matter in the gaseous state, for in this form 
substances obey laws which in point of simplicity and extensive 
application are not approached by substances in either of the other 
states of aggregation. We have the simple laws of Boyle, G-ay- 
Lussac, and Avogadro, which connect in a perfectly definite manner 
the pressure, temperature, volume, and number of molecules in all 
gaseous substances, whatever their chemical nature or other physical 
properties may be. These laws point to great simplicity in the ' 
mechanical structure of gases, and to the sameness of this structure 
for all gases. Various hypotheses have from time to time been put 
forward to explain the behaviour of gases, but only one has been 
found to be at all satisfactory, and to some extent applicable to the 
other states of matter. 

This hypothesis is called the kinetic theory of gases, and is, in its 
present form, chiefly due to the labours of Clausius and Maxwell. 
According to this theory, the particles of a gas — which are identical 
with the chemical molecules — are practically independent of each 
other, and are briskly moving in all directions in straight lines. It 
frequently happens that the particles encounter each other, and also 
the walls^ of the vessel containing them ; but as both they and the 
walls are supposed to behave like perfectly elastic bodies, there is no 
loss of their energy of motion in such encounters, merely their 
directions and relative velocities being changed by the collision. 


The total pressure exerted by a gas on the walls of the vessel con- 
taining it is due to the impacts of the gas molecules on these walls, and 
is measured by the change of momentum experienced by the particles 
on striking the walls. Suppose a particle of mass m to be moving 
with the velocity c, and let it impinge on the wall at right angles. 
The particle will rebound in its line of approach with a velocity equal 
to its original velocity, but of course with the opposite sign. The 
original momentum was rac, the momentum after collision is - mc, so that 
the change of momentum is 2mc. If we calculate now the change of 
momentum suffered by all the particles of a quantity of gas in a given 
time by collision with the walls, we are in a position to give the total 
effect on the walls, and thus the pressure. For the sake of simplicity, 
we imagine that the vessel is a cube, the length of whose side is s, and 
that all the molecules have the same mass m, and the same velocity c. 
Let the total number of molecules in the gas be n. The molecules, 
according to the original assumption, are moving in all directions, but 
the velocity of each may be resolved into three components parallel to 
the edges of the cube, the components being related to the actual 
velocity by means of the equation x 2 + y 2 + z 2 - c 2 . Consider a single 
molecule with respect to its motion between two opposite sides of the 
cube. Its velocity component in this direction is x, and the number of 

impacts on the sides in unit time will be -. The change of momentum 

on each impact is 2mx, so that in unit time the total change of momentum 

of the molecule caused by impacts on the walls considered is 2mx. , 


The action of the molecule on these walls, therefore, is , and on 


2 7)%v 2 7/iz 

the other two pairs of walls it will be — - , and . Thus the 

r s s 

action of the molecule on all the walls in unit time is 

2m(x 2 + f + .? 2 ) _ 2mc 2 
s s 

Now in the whole quantity of gas there are n molecules, so 

that the total action of the gas on the walls of the cube is * 

The surface of the six sides of the cube is 6s 2 , and the quotient 

-Try therefore gives the action per unit surface. But s 3 is equal to i\ 


the volume of the cube, so that we have finally p= -5 — or pv = \nmc 2 . 


All the magnitudes on the right of this equation are constant at constant 
temperature, hence the product of the pressure and volume of the gas is 
constant, and thus from the assumptions of the kinetic theory we deduce 
Boyle's Law. In the above deduction the vessel was supposed to have the 


cubical form ; but any space may be considered as made up of a large 
number of small cubes, and as the impacts on the opposite sides of all 
faces common to two cubes would exactly neutralise each other, the 
presence of these internal partitions does not affect the impacts on 
the outer faces of the external cubes, which in the limit constitute 
the walls of the containing vessel. 

Since Jwc 2 is the kinetic energy of a single molecule, the expres- 
sion ^nmc 2 or f n . %mc 2 may be read as two-thirds of the total kinetic 
energy of the gas, and we may say that the product of the pressure 
and volume of a gas is equal to two-thirds the kinetic energy of its 
molecules. We learn that systems of moving particles, such as gases 
are imagined to be on the assumptions of the kinetic theory, are in 
equilibrium with each other when the kinetic energies of their particles 
are equal ; and we know that gases are actually in physical equilibrium 
when their pressures and temperatures are equal, i.e. they may then be 
mixed without the pressure or temperature undergoing alteration. Let 
us consider a number of gases at the same pressure and at the same 
temperature. If the temperature of the gases is in every case altered 
in the same degree, the pressure remaining constant, the gases are still 
in physical equilibrium, and consequently the kinetic energies of their 
particles must have altered in an equal degree. But the product of 
pressure arid volume of a gas is proportional to the kinetic energy of 
its particles, and this product has therefore been altered to the same 
extent for each gas. Since, however, the pressure remained constant 
throughout, the volume of each gas has thus undergone |fe?igame rela- 
tive change. Thus the kinetic theory enables us to deduce that the 
volume of different gases is affected equally by the same change of 
temperature if the pressure remains constant, i.e. that all gases have 
the same coefficient of expansion. 

Avogadro's Law may also be deduced from the kinetic theory by 
making use of considerations similar to the above. Take equal volumes 
of two gases at the same temperature and pressure. Since p=p\ and 
v = v', pv =pv, and consequently 

n . \mc L — %n r . \r 

But the kinetic energies of the two gases must also be equal, since they 
are in mechanical equilibrium, i.e. 

±mc 2 = %m'c' 2 , 

whence, dividing the first equation by the second, n = n. Equal 
volumes of different gases, therefore, at the same temperature and pres- 
sure contain the same number of molecules. 

I Spy 

We may write the equation pv = ^nmc 2 in the form c — k /^£-, or 
since — is the density of the gas, being its weight divided by its 


22,380 cc, so that c =^ / ^— - = 46,100 centi- 


volume, c = I —-. It appears, then, that the speed of the molecule of 

a gas is inversely proportional to the square root of the density of the 

gas, a result which is in harmony with the experimental result that the 

velocity of diffusion (and the velocity of transpiration) of a gas is 

inversely proportional to the square root of its density. 

It is possible to obtain a value for the speed of the molecules of 

a gas by substituting the known values in the equation for the velocity ^*> 

given above. Thus for 32 g. of oxygen under standard conditions^* 

we havejp= 1,013,000 dynes 1 per square centimetre, mn = 32 g., and 

/3x 1,013,000 x 22,380 


metres per second. The molecule of oxygen therefore at 0° C. moves at the 

rate of 46,100 cm. per second, or nearly 18 miles per minute. The speed 

of the molecules of any other gas at any temperature may be got from the 

/ d T 
formula c a = c / -= — ^— - in which d 1 is the density of the gas, d the 

density of oxygen, and T the temperature in the absolute scale. 

In the foregoing we have spoken of the velocity of the particles of 
a gas as if all the particles had the same velocity. This, however, can- 
not be the case, for even though the particles had the same velocity at 
the beginning of any time considered, the velocities of the individual 
particles would speedily assume different values owing to their encoun- 
ters. It must be understood, therefore, that the velocity in the above 
formulae means a certain average velocity, some of the particles having 
a greater and some a smaller speed than corresponds to this value. 
The bulk of the particles have velocities in the neighbourhood of this 
mean velocity, and the farther we diverge from the mean, the fewer 
particles we find possessing the divergent values. 

If two different gases are brought together, their particles in virtue 
of their rapid motion in straight lines will soon leave their fellows and 
mix with the particles of the other gas. This process of intermixture 
we are acquainted with practically as gaseous diffusion. Two gases, 
no matter how different their densities may be, will mix uniformly if 
brought into the same space, but the rate of intermixture is very much 
slower than what we should expect from the rate at which the particles 
move. This discrepancy, however, may be easily explained. The 
particles in a gas at ordinary pressure are comparatively close together, 
and consequently encounter each other frequently; so that, though 
their rate of motion between individual encounters is very great, their 
path between points any distance apart is, owing to these encounters, a 
very long and irregular one, and the rate of mixing is therefore com- 
paratively small. After the mixture of two gases has attained the 
same composition in every part, there is no further apparent change ; 

1 One gram weight is equal to 981 dynes. 


"but the motion of the particles, and thus the mixing process, is sup- 
posed to go on as before, only now the further mixing does not alter 
the composition. 

The kinetic theory may be applied in a general way to the study of 
the processes of evaporation and condensation. In a liquid the par- 
ticles have not the same independence and free path as gas particles, 
although they are in general identical with the gaseous molecules and 
have equal velocities. A gaseous substance, in virtue of the freedom of 
its molecules, can expand so as to fill any space offered to it. A liquid 
does not do so at low pressures, but retains its own proper volume, al- 
though its molecules still possess sufficient independence to move easily 
between collisions, and thus enable the liquid under the influence of 
gravitation to accommodate itself to the shape of the vessel containing 
it. In spite of the clinging together of the liquid molecules, it happens 
that some of them near the surface have sufficient motion to free them- 
selves from their neighbours, and leaving the liquid altogether, to be- 
come free gas molecules. If these gas molecules move away unhindered, 
other molecules from the liquid will take their place; and so the 
liquid will go on giving off gas molecules until all has evaporated. 
If, however, the liquid is kept in a closed space, the gas molecules 
which leave its surface will be able to proceed no farther than the 
walls of this space, and must eventually return in the direction of the 
liquid. It will consequently happen that some of them will strike the 
surface of the liquid again and be retained by it. But the liquid 
molecules still continue as before to become gas molecules and leave 
the surface of the liquid, so that at one and the same time there are 
molecules entering and molecules leaving this surface. When in^a 
given time as many molecules leave the liquid as are reabsorbed by it, 
no further apparent change takes place — the relative quantities of the 
liquid and the vapour remain the same. A stationary state of balance 
or equilibrium has thus set in, and we may now look at what 
determines this state. The number of molecules leaving the liquid 
depends on the temperature, for it is only those molecules which attain 
a certain velocity that will succeed in freeing themselves; and the 
motion of the molecules of a liquid, like those of a gas, depends 
directly on the temperature. The number of the molecules reabsorbed 
by the liquid depends on the number of gas molecules striking the 
surface in a given time, i.e. on the number of molecules contained in 
a given space and on their speed. As we have seen, this number and 
the speed together determine the pressure exerted by a gas, so the 
number of molecules reabsorbed depends on the pressure. Temperature 
thus regulates the number of molecules freed, and gaseous pressure 
the number of molecules bound in a given time ; consequently for each 
state of equilibrium when these two numbers are equal, a definite 
temperature will correspond to a definite gaseous pressure of the 
vapour in contact with the liquid — or vapour pressure of the liquid, 


as it is shortly termed. Every liquid, therefore, has at each tempera- 
ture a definite vapour pressure ; and this vapour pressure increases as 
the temperature rises, for more molecules at the high temperature will 
have the speed necessary to free them. It may be noted that as it 
is the molecules with the greatest speed, i.e. with the highest 
temperature, that first leave the liquid, the average temperature of the 
liquid must sink as evaporation goes on, unless heat is supplied from 
an external source. 

The kinetic theory not only gives us the ordinary gas laws, which 
are, strictly speaking, obeyed only by ideal gases and not by any 
actual gas, but also when properly applied affords us a probable 
explanation of the deviations from the gas laws which are experi- 
mentally found. So far, we have considered the gas molecules as 
mere physical points occupying no volume whatever ; but certainly if 
gaseous particles are supposed to exist at all, they must be supposed to 
possess finite though small dimensions. It is evident that the volume 
in which these particles have to move is not the volume occupied 
by the whole gas, but this volume minus at least the volume of the 
particles. So long as the volume occupied by the gas is great and 
the pressure small, the volume of the particles vanishes in comparison 
with the total volume, and the gas laws are closely followed; but 
when the pressure is great and the total volume small, the volume of 
the particles themselves bears a considerable proportion to this whole, 
with the consequence that the divergence from the gas laws is great. 
Owing to this cause, the pressure would increase in a greater ratio 
than the volume would diminish, as the following reasoning will serve 
to show. Suppose a molecule to be oscillating between two parallel 
walls in a direction at right angles to them, and suppose the distance 
between the walls to be equal to 100 times the diameter of the 
molecule. It is evident that the molecule from its contact with one 
wall has to travel, not 100 diameters before it comes in contact with 
the other, as it would have to do if it were a point without sensible 
dimensions, but only 99. It will therefore hit the walls oftener in a 
given time than if it were without sensible dimensions, and that in 
the ratio of 100 to 99. Now suppose the distance between the walls 
to be reduced to 10 molecular diameters. The particle has now only 
to travel 9 times its own diameter in order to pass from contact with 
the one wall to contact with the other. It will therefore in a given 
time hit the walls oftener in the ratio of 10 to 9, or 100 to 90, than 
if it were a mere point. By diminishing the distance of the walls to 
one-tenth, therefore, we have increased the pressure not to ten times 
the original value, but to this value multiplied by 99 : 90, i.e. the 
pressure has increased to eleven times its former magnitude. 

We might now write the gas equation j>v = BT in the form 
p(v - b) = BT, 
where b is a constant for each gas depending on the magnitude of the 


molecules of the gas. But there is still another influence at work 
which interferes with simple obedience to the ideal gas laws. The 
particles of a liquid undoubtedly exercise a certain attraction on each 
other, and this attraction must still persist when the liquid particles 
have become particles of vapour, only in the latter case the particles 
are in general so far apart that the effect of the attraction is incon- 
siderable. If the gas is compressed into a smaller volume, however, 
the influence becomes felt more decidedly owing to the comparative 
proximity of the particles. Van der Waals has assumed that this 
attraction is proportional to the square of the density of the gas, or 
reciprocally proportional to the square of the volume. The effect of 
the mutual attraction of the particles is the same as if an additional 
pressure were put upon the gas, so that the correction is applied by 
adding it to the value of the external pressure. If a denotes the 
coefficient of attraction, i.e. the value of the attraction when the gas 
occupies unit volume, then the correction for any other volume 

is -g. The equation for the behaviour of a gas under all conditions 

is, therefore, according to van der "Waals, 

(p + £)(v-b) = ]iT. \ 

This equation not only gives the, behaviour of the so-called permanent 
gases very accurately up to high pressures, but even that of a 
comparatively compressible gas like ethylene. The following table 
gives the values of pv for ethylene actually found by Amagat, and 
those calculated from the equation 

P + °' Q °f 86 ) ( v - 0*0024) = constant, 

p being expressed in atmospheres, and v being taken equal to 1000» 
when^>= 1. 




The agreement between the observed and calculated values is very 
satisfactory. If the gas were an ideal gas the values of pv would 
remain the same for all pressures. We see, however, that this 
constancy is far from being fulfilled. The gas is at first more 
compressible than corresponds to Boyle's Law, and then at higher 



















pressures less compressible, the minimum value of the product 
occurring when the pressure is about 80 atmospheres. From the 
form of the equation it may be seen that the two corrections act in 
opposite ways, the value of the product pv being increased by the 
attraction, and diminished by the finite dimensions of the molecules. 
At low pressures the effect of the attraction greatly overweighs the 
volume correction which in its turn becomes preponderant when the 
pressure reaches a high value and the total volume becomes small. 
With ethylene at the temperature considered, the two corrections 
balance each other at about 80 atmospheres, and here the gas within 
narrow limits of pressure obeys Boyle's Law, for the product pv then 
remains sensibly constant. 

All gases hitherto investigated, with the single exception of 
hydrogen, give similar deviations from Boyle's Law : the product of 
pressure and volume at first diminishes, afterwards to increase as the 
pressure rises. At higher temperatures the deviations are. of the same 
kind, but not so marked. This may be seen directly from the formula, 
the constants a and b being independent of the temperature, and the 
value of the expression on the right-hand side increasing in direct 
proportionality with the absolute temperature. In the case of 
hydrogen there is no preliminary diminution of the value of pv, on 
account of the constant of attraction a being so small that its effect is 
counterbalanced from the first by the effect of the constant b. 

The equation of van der Waals is especially interesting in its 
application to the continuous passage from the gaseous to the 
liquid state, as it holds good not only for gases, but also in many 
' ways for liquids. If we rearrange the equation so as to give co- 
efficients of the powers of v, we obtain 

v s 

ET\ 9 a ab 

- [o+ — )v z + ~v = 0, 

\ p J p p 

This cubic equation has in general three solutions, so that for each 
value of p we have in general three corresponding values of v. The 
graph of the equation for constant values of a and b, and for various 
values of T, is given in Fig. 15. The isothermal curves thus obtained 
would represent the behaviour of a substance at various temperatures. 
The curves for the lower temperatures are wavy in form, and are cut 
by horizontal lines of constant pressure, sometimes in three points and 
sometimes in one. When the curve is cut by the horizontal line only 
once, the point of intersection gives the real solution of the equation, 
the other two solutions being imaginary. The resemblance of these 
curves to the curves on p. 77, which roughly express the result of actual 
experiment, is at once evident. In the case of the theoretical curves 
we have no sudden breaks such as we have in the actual discontinuous 
passage of vapour into liquid by increasing pressure. Van der Waals' s 
equation assumes a continuous passage from the liquid to the vaporous 



state, and vice versa, such as we find when we take the temperature and 
pressure above their critical values for the substance under considera- 
tion. When the passage be- 
tween the two states is dis- 
continuous, as it usually is, 
we proceed along a horizontal 
line from one part of the 
theoretical curve to another, 
this line of constant pressure 
cutting the curve in three 
points. The least of the 
volumes corresponding to 
the constant pressure is the 
volume of the substance as- 
liquid; another volume, the 
greatest, is the volume of 
the vapour derived from the 
liquid; the substance in a 
homogeneous state occupying 
the third intermediate volume 
is unknown. By studying 
supersaturated vapours and 
superheated liquids, we can 
advance along the theoretical 
curve for short distances 
beyond B and C without 
discontinuity, but the sub- 
stance in these states is com- 
fig. is. paratively unstable. In the 

neighbourhood of the third 
volume the state of the substance is essentially unstable, increase of 
pressure being followed by increase of volume, and so we cannot hope to 
realise it. Van der Waals has pointed out, however, that in the surface 
layer of a liquid, where we have the peculiar phenomena of surface 
tension, it is possible that such unstable states exist, and that the 
passage from liquid to vapour may after all in the surface layer be 
really a continuous one. 

It will be noticed that as the temperature increases, the wavy 
portion of the curve gets continually smaller, and the three volumes 
get closer and closer together, finally to coalesce in a single point. 
Here the three solutions of the equation become identical, the volume of 
the liquid becomes equal to the volume of the substance as gas, and 
there is no longer any discontinuity or distinction between the liquid and 
gaseous states. In short, the substance at this point is in the critical 
condition : the curve is the curve of the critical temperature, the 
pressure is the critical pressure, and the volume is the critical volume. 


When the three roots of a cubic equation become equal, certain 
relations exist between this triple root and the coefficients of the 
powers of the variable, If the equation is 

x 3 -Ax 2 + Bx-C=0, 

m& if the triple root is represented by £, the following relations hold 
good : — 

3^A; 3? = B; £*=C. 

In van der Waals's equation there are only the pressure and 
volume of the gas, the constants a and b, and the gas constant B. 
Now we can express the constant R in terms of the constants a and b 
as follows. Under normal conditions, let jp = 1,^=1, and T = 273. 
Then the equation 


(l+a)(l-b) = 273R 

(l+q)(l-6) . 
"~ 273 

or, if we make -^-^=z/3 y the coefficient of expansion, we have 

11 = /3(1 + a)(l - b). 

Van der Waals's equation then becomes (cp. p. 91). 

W 6 + /?r(i + q)(i-5)y +V g* =a 

V p J p p 

If we denote now by <£, 7r, and 6 the critical values of v, p, and T 
respectively, we obtain for the critical equation, in which the three 
roots become identical, 

3<£ = 

= b + 


+ a) 



3<j> 2 = 

= -j 

4> 3 = 

= ; 



<£ = 36 (critical volume), 
it = r^jo (critical pressure), 

^ 27^(1 Ta)(l-b ) (° ritical tem P eratoe >- 


Here we have general expressions for the critical values of a substance 
in terms of the constants which express the deviation of the substance 
from the laws for ideal gases ; and conversely we can give the 
numerical values of these constants from the values of the critical 
data, viz. — 


These relations have been tested in several cases, and a fair approxi- 
mation of the calculated to the observed values has been found. 

If in van der Waals's equation we express the values of the pressure, 
temperature, and volume as fractions of the corresponding critical 
values, and at the same time express these latter in terms of the' 
deviation constants, the equation becomes 

(71 \ 
€ + --J(3»-l) = 8m> 

in which € = -; n = - ; ra = ~. 

Here everything connected with the individual nature of the substance 
has disappeared, and we have an equation which applies under certain 
restrictions to all substances in the liquid or gaseous state, just as the 
gas equation holds good for all gases, independently of their specific 

The chief point to be noted here is that whereas for gases the 
temperature, pressure, and volume may be measured in the ordinary 
units without impairing the validity of the comparison of different 
gases, it is necessary in the case of liquids to effect the comparison 
under "corresponding " conditions, the temperatures of the two liquids 
to be compared being, for instance, not equal on the thermometric scale, 
but being equal fractions of the critical temperatures of the two 

It is one of the tasks of physical chemistry to compare the physical 
properties of different substances, and to trace, if possible, some 
connection between their magnitude and the chemical constitution 
of the substances considered. Now we know that most physical 
properties of substances vary with the temperature and pressure. 
The question therefore arises : At what temperature and pressure are 
we to compare the properties of different substances 1 It is evidently 
purely arbitrary to make the comparison at the so-called standard 
conditions of 0° and 760 mm., for these conditions have no relation 
whatever to the properties of the substances themselves, and are merely 
chosen for convenience sake as being easily attainable in the circum- 
stances in which we work. Yan der Waals answers the question by 


saying that the properties ought to be compared at corresponding 

temperatures and pressures, meaning thereby at temperatures and 

pressures which are equal fractions of the critical values in the absolute 

scale. Suppose, for example, we were to compare ether and alcohol 

with respect to some particular property. The critical temperature of 

ether is 194° C, or 467 absolute; that of alcohol is 243° C, or 516 

absolute. Let the property for the alcohol be measured at 60° C, 

then we shall have as the corresponding temperature x for ether 

273 + a; 273 + 60 OQ0 n _ , „ , 

— = — —ft — or ic = 28 G. lne pressure when small has not, as 

a rule, a great effect on the properties of liquids, so that in general we may 
make the comparison at the atmospheric pressure without committing 
any serious error. It should, however, be stated at once that the data for 
the comparison of different substances under corresponding conditions 
are for the most part still wanting, so that it is not known whether the 
theoretical conditions would lead to sensibly greater regularities than 
those observed among the properties when measured under more usual 

The kinetic theory affords also some account of the phenomena 
of solution. If we take, for example, the case of the solution of 
a gas in a liquid, we can easily see that the gas molecules impinging 
on the surface of the liquid may be held there by the attraction of 
the molecules of the solvent. When, however, a number of the gas 
molecules have accumulated in the liquid, some of them, in virtue of 
their motion, will fly out from the surface of the solution, and this 
will happen the more frequently the more molecules there are 
dissolved in the liquid. But as the number of gas molecules striking 
the surface of the liquid remains constant at constant pressure, it 
will at last come to pass that the number of molecules entering and 
leaving the liquid will be the same. There is then equilibrium, and 
the liquid is saturated with the gas. As the number of gas molecules 
striking the liquid surface is proportional to the pressure, the number 
of molecules leaving that surface when the liquid is saturated, and 
consequently the number of molecules dissolved in the liquid, is 
likewise proportional to the pressure. This is Henry's Law, and 
Dalton's Law also follows at once ; for in a gaseous mixture the 
number of molecules of each gas striking the surface is proportional 
to the partial pressure in the mixture, and independent of the other 
components. It will be seen from this explanation that there is a 
great similarity between the solution of a gas in a liquid and the 
phenomena of evaporation and condensation. 

The same analogy appears when we consider the solution of a 
solid. When a soluble crystalline substance is introduced into a 
solvent, some of its particles become detached and enter the solvent. 
After a time certain of these detached particles come into contact 
with the solid again, and are retained by it. This give-and-take 


process goes on until the same number of particles leave the solid and 
return to it in a given time. No further apparent change then takes 
platfe, and the solution is saturated. The number of particles which 
return to the solid evidently depends on the number of them in unit 
volume of the solution, i.e. on the strength or concentration of the 
solution. If the solid is brought into contact with a stronger solution 
than the above, more particles will enter the crystal than will leave 
it, and so the crystal will increase in size. Such a solution is 
supersaturated with regard to the solid. In a weaker solution, fewer 
particles will come into contact with the solid and be retained by it 
than will leave it, i.e. the solution is unsaturated and the crystal will 
dissolve, in part at least. 

In the chapter on evaporation and condensation we had occasion 
to refer to the vapour tension of liquids, meaning thereby the tendency 
of the liquids to pass into vapour under the specified conditions. 
There is equilibrium when the vapour tension of the liquid is balanced 
by the gaseous pressure of the vapour above the liquid. A similar 
term has been employed to express the tendency of a substance to 
pass into solution, the substance having a definite solution tension 
for each solvent it is brought into contact with. When the pressure 
of the dissolved substance in the solution is equal to the solution 
tension of the solid there is equilibrium. Here a new conception is 
introduced, namely, the pressure of a substance in solution. What 
this pressure is, and how it may be measured, will be seen in 
Chapter XVI. 

A brief account of the Kinetic Theory will be found in 
Clerk-Maxwell, Theory of Heat, chap. xxii. 

The student is also recommended to read, in connection with this chapter, 
J. P. Kuenen, on " Condensation and Critical Phenomena," Science Pro- 
gressy New Series, 1897, vol. i. p. 202 and p. 258. 



A substance is in general capable of existing in more than one 
modification. For example, water may exist as ice, as liquid water, 
or as water vapour. Sulphur exists as vapour, liquid, and as two 
distinct solids, namely, as monoclinic and as rhombic sulphur. Para- 
azoxyanisoil, as we have seen, forms not only solid and gaseous 
modifications, but can also exist as a crystalline liquid distinct from the 
ordinary non- crystalline liquid. All such modifications, when they 
exist together, are mechanically separable from each other, and are in 
this connection called phases. A single substance may assume the form 
of many different phases, but these phases cannot in general all exist 
together in stable equilibrium, being subject to certain restrictions regulat- 
ing their coexistence, which may be stated in the form of definite rules. 
As a familiar example we shall take the substance water in the 
three phases — ice, water, and vapour. The physical conditions 
determining the equilibrium of these phases are temperature and 
pressure. We know that at the pressure of 1 atmosphere, water 
is in equilibrium with ice at the temperature of zero centigrade, and 
with water vapour at the temperature of 100° centigrade. For a 

j given pressure, then, there is a definite temperature of equilibrium 
between such a pair of phases ; and we shall also find that for a definite 
temperature there is a definite equilibrium pressure. Consider the 

?two phases, water and water vapour. To each temperature there 
corresponds a fixed vapour pressure, which is the pressure of water 

- vapour, or the gaseous phase, which is in equilibrium with the water 
or liquid phase. By drawing the pressure - temperature diagram, 
therefore, of a substance, we are enabled to study conveniently the 
equilibrium between its phases. In Fig. 16 the line OA represents 
the vapour-pressure curve of water, each point on the line correspond- 
ing to a certain pressure measured on the vertical axis, and to a certain 
temperature measured on the horizontal axis. For the sake of 
clearness, the curves in the diagram have all been drawn as straight 

'lines. Ice, like water, has a vapour-pressure curve of its own, and 




this has been represented in the diagram by the line OB. It will be 
observed that the two vapour -pressure curves have not been repre- 
sented as one continuous line, but as two lines intersecting at a point 
0, If we inquire into the meaning of this intersection, as interpreted 
from the diagram, we find that at a certain temperature t ice and water 
have the same vapour pressure, for the point corresponding to this 
temperature belongs both to the vapour-pressure curve of water and 
the vapour-pressure curve of ice. It is easy now to show that there 
is in fact a temperature at which the vapour pressures of ice and water 
are identical. Water at its freezing point is in equilibrium with ice, 
i.e. ice and water can coexist at this temperature in any proportions 

Fia. 16. 

whatever, and these proportions will remain unchanged if the mixture 
is surrounded only by objects of this same temperature. Now let the 
water and ice coexist, not, as is usual, under the pressure of one 
atmosphere, but under the pressure of their own vapour. The 
temperature of coexistence will no longer be exactly 0°, but a 
temperature very slightly higher; otherwise the conditions are 
unchanged. If the ice, at this equilibrium temperature between ice 
and water, have a vapour pressure higher than that of water at the 
same temperature, diffusion will tend to bring about equalisation of 
pressure, i.e. the pressure of vapour over the ice will become less than 
its own vapour pressure, so that ice will evaporate ; and the pressure 
of vapour over the water will become greater than its own vapour 
pressure, so that water will be formed by condensation. Ice will have 
therefore been converted into water, which is contrary to our original 


assumption that the proportions of water and ice present are not under 
the circumstances subject to alteration. Similarly, if water at the 
equilibrium temperature have a greater vapour pressure than ice, 
ice would be formed indirectly through the vapour phase at the 
expense of the liquid water, so that our assumption in this case also 
would be contradicted. There only remains, then, the alternative 
that the vapour pressures of ice and water are equal when the ice 
and water are in equilibrium, which is in accordance with the 
representation of the diagram. At any point on the line OA water 
and water vapour can coexist in equilibrium, at any point on the line 
OB ice and water vapour can coexist. At the point 0, where these 
two lines intersect, all three phases can exist together in equilibrium, 
and such a point is therefore called a triple point. When a substance 
can only exist in three phases, only one triple point is possible. The 
triple point in the case of water is not quite identical with the melting 
point of ice, because the melting point is, strictly speaking, defined as the 
temperature at which the solid and liquid are in equilibrium when the 
pressure upon them is equal to one atmosphere. At the triple point the 
pressure is not equal to the atmospheric pressure of 760 mm. but to the 
vapour pressure of ice or water, which is only 4 mm. Now, it has been 
shown, both theoretically (cp. Chap. XXVII.) and experimentally, that 
pressure lowers the equilibrium temperature of ice and water by about 
0*007° per atmosphere, so that the freezing point under atmospheric 
pressure is about 0*007° lower than the triple point. The effect of 
pressure on the melting point of ice may be represented in the 
diagram by the line OG, inclined from the triple point towards the 
pressure axis. At any point on this line, ice and water are in 
equilibrium with each other, the temperature of equilibrium falling 
with increase of pressure. The diagram for equilibrium of the three 
phases of water consists therefore of three curves meeting in a point, 
the triple point. At any other point on the curves, two phases can 
coexist in equilibrium : — 

(a) Water and water vapour on OA ; 

(b) Ice and water vapour on OB ; 

(c) Water and ice on OC. 

At 0, the common point of intersection, all three phases are in 
equilibrium together. The three lines divide the whole field of the 
diagram into three regions. At pressures and temperatures represented 
by any point in the region AOB water can only exist permanently in 
the state of vapour. At any point in the region AOC it can only 
exist as liquid water, and at any point in the region BOC it only 
exists as ice. The curve OA separates the region of liquid from the 
region of vapour, but the separation is not complete. The curve is 
the curve of vapour pressures, and, as we have already seen, there is 
a limiting pressure beyond which the vapour pressure of a liquid 


cannot rise. This is the critical pressure of the substance, and it is 
attained at the critical temperature. The curve OA, therefore, ceases 
abruptly at a point A, the values of the pressure and temperature at 
which are the critical values. Beyond A there is no distinction 
between liquid and vapour ; the two phases have become identical. 

It is possible to pass from a point M in the liquid region to a point 
N in the gaseous region in an infinite number of ways, which may be 
represented on the diagram by straight or curved lines. If these lines 
cut the line OA, there is discontinuity in the passage, for at the pressure 
and temperature represented by the point of intersection the two phases 
will coexist. For example, we may pass from M to N, by means of 
lines parallel to the axes, along the path MLN. The line ML represents 
increase of temperature at constant pressure ; the line LN diminution 
of pressure at constant temperature. The pressure at L is greater 
than the vapour pressure of the liquid at the constant temperature 
considered, and so the substance exists at this point as liquid only. 
As the pressure is gradually released, a point is at last reached at 
which it is exactly equal to the vapour pressure of the liquid. The 
liquid now begins to evaporate, and the two phases exist together, 
the point at which this occurs being the point of intersection of LN 
with OA. No further reduction of pressure can be effected until all 
the liquid has been converted into vapour, after which the pressure 
may be diminished until it attains the value represented by the point 
N. If, on the other hand, we follow the line MPQN, which does not 
cut the line OA, we can pass from the state of liquid at M to the 
state of vapour at N without any discontinuity whatever. We first 
increase the temperature, following the line MLP, to a value above the 
critical temperature, the pressure being all the time above the critical 
value. This takes us into the region where there is no distinction 
between liquid and vapour, so that by first reducing the pressure and 
then^ lowering the temperature we pass without any break to a sub- 
stance in the truly vaporous state at N, the substance at no time having 
been in the state of two distinct phases. 

In what has been said above as to the condition of the substance 
in the various regions of the diagram, it has been assumed that stable 
states only are under consideration. If we disregard this restriction, 
then we may have, for example, liquid water in the region BOC, for 
water may easily be cooled below its freezing point without actually 
freezing, and exist as liquid at points to the left of 0. Such super- 
cooled water has a vapour pressure curve which is the continuation of 
the curve OA, and has been represented in the diagram by the dotted 
line OA'. This curve lies above the vapour-pressure curve for ice, 
so that at any given temperature below the freezing point the vapour 
pressure of the supercooled liquid is greater than the vapour pressure 
of the solid. This rule holds good for all substances, and we find' in 
general that the vapour pressure of the stable phase is less than t the 

xi THE PHASE 1 ft tJJtf? > ' ^ ^ * Jj * i A 1 1 

vapour pressure of the unstable phase. It should be noted that the 
instability in such examples is only relative. A supercooled liquid 
may be kept for a very long time without any solid appearing 
(cp. Chap. VIII.), but as soon as the smallest particle of the sub- 
stance in the more stable solid phase is introduced, the less stable, 
or, as it has been called, the metastable phase is transformed into 
it. That the metastable substance should have a higher vapour 
pressure than the stable substance is not surprising, if we consider 
that the phase of higher vapour pressure will always tend to pass 
into the phase of lower vapour pressure when the two substances 
are allowed to evaporate into the same space, although they are 
not themselves in contact. The vapour of the substance of higher 
vapour pressure will on account of that higher pressure diffuse 
towards the substance of lower vapour pressure and there condense. 
More of the metastable phase will then evaporate in order to 
restore equilibrium between itself and the vapour, with the result 
that there will again be diffusion and condensation on the stable 
phase until all the metastable phase has been thus indirectly 
converted by evaporation into the stable phase. The vapour at 
pressures and temperatures represented by points on the line OA' is 
in an unstable state with regard to the solid ice, being supersaturated, 
although it is only saturated with regard to the supercooled liquid. It 
is likewise possible to supersaturate vapour at temperatures above the 
freezing point, i.e. to have the substance in the state of vapour in the 
region COA ; and also to have a liquid substance in the region AOt 
by superheating. Water, for example, if free from dissolved gases, 
may be heated to 200° or over at the atmospheric pressure without 
boiling. It has always been found impossible, on the other hand, to 
heat a solid above its melting point. Water in the form of ice has 
never been observed in the region COA. 

We shall next proceed to the consideration of a substance capable 
of existence in more than three phases, taking sulphur as our ex- 
ample. Here we have not only the liquid and vaporous phases, but 
the two solid phases of rhombic and monoclinic sulphur. Khombic 
sulphur is the crystalline modification usually met with, and this on 
heating rapidly melts at 115°. If we keep it at a temperature of 
100°, however, for a considerable time, we find that it becomes 
converted into the other modification, monoclinic sulphur. This latter 
on heating does not melt at 115° but at 120°, in accordance with the 
general rule that each crystalline modification of a substance has its 
own melting point. If the monoclinic sulphur be cooled to the 
ordinary temperature, it gradually passes again into the rhombic 
modification. We should be inclined, therefore, to say that at the 
ordinary temperature rhombic sulphur is in a stable state, while 
monoclinic sulphur is in a metastable condition. At 100° the reverse 
is the case : here monoclinic sulphur is the stable variety, and rhombic 



sulphur the metastable variety. We have seen that in the case of 
solid and liquid there is a temperature at which both phases are stable 
together, namely, the melting point. Above or below this temperature 
only one of the phases is stable. We should therefore expect by 
analogy that there is a temperature at which the two solid phases of 
sulphur should be equally stable, ie. should be able to coexist without 
any tendency of the one to be converted into the other. Careful 
experiment has revealed such a temperature. At 95*6°, the 
transition or inversion temperature, both rhombic and monoclinic 
sulphur are stable, and can exist either separately or mixed together in 


115 120 

Fig. 17. 1 


any proportions. Below this temperature the monoclinic phase gradually 
passes into the rhombic phase ; above it, the rhombic phase gradually 
passes into the monoclinic. A transition temperature of this sort is 
then quite comparable to a melting point, the chief difference being 
that while a solid can never be heated above its melting point without 
actually fusing, a substance like rhombic sulphur may be heated above 
its transition point without undergoing transformation. It is thus 
possible to investigate the properties of rhombic sulphur up to its 
melting point, 115°, although between 95*6° and that temperature it is 
in a metastable condition, and is apt to suffer transformation into 

1 The point C in the actual diagram would lie very much higher than is here 


the stable monoclinic modification. The transition point, like the 
melting point, is affected by pressure, and in the case of sulphur 
increase of pressure has the effect of raising the transition temperature. 
All these phenomena may be represented diagrammatically by means of 
temperature-pressure curves (Fig. 17). The line OB in the figure 
represents the vapour-pressure curve of rhombic sulphur ; OA is the 
vapour pressure curve of monoclinic sulphur. These vapour pressure 
curves must meet at the transition point, for at that temperature both 
modifications are equally stable and must have the same vapour 
pressure. Below that temperature the vapour pressure of the 
metastable monoclinic phase must be greater than that of the stable 
rhombic phase. The line OA', therefore, which is the prolongation of 
the line OA, represents the vapour-pressure curve of monoclinic sulphur 
below the transition point. Above 95*6° rhombic sulphur is the 
metastable phase, and consequently has the greater vapour pressure. 
This is represented in the diagram by the dotted line OB', which is 
the continuation of the line OB. The line OC gives the effect of 
pressure on the transition point, sloping upwards away from the 
pressure axis in order to represent rise of transition point with rise of 
pressure. The point thus corresponds very closely to the point 
of Fig. 16, being like it a triple point at which there is stable 
equilibrium of three phases, viz. rhombic, monoclinic, and gaseous. 
The lines OA, OB, and OC diverging from represent as before the 
conditions of equilibrium between pairs of phases, and the dotted 
lines OA' and OB 7 similar conditions in metastable regions. 

It has been already stated that monoclinic sulphur melts at 1 20°. 
At this point also a triple point must exist, for here monoclinic 
sulphur, liquid sulphur, and sulphur vapour are in equilibrium. The 
melting point of monoclinic sulphur is raised by pressure, instead of 
being lowered, as is the case with water. We have therefore three 
curves intersecting at A, namely, OA, representing the vapour pressure 
of monoclinic sulphur ; AD, representing the vapour pressure of liquid 
sulphur ; and AC, representing the influence of pressure on the melting 
point. It happens that the lines OC and AC, representing the effect 
of pressure on the transition point and on the melting point of 
monoclinic sulphur, although both sloping upwards from the pressure 
axis, meet at a point C, corresponding to a temperature of 131°. 
This point C is also a triple point, for at the pressure and temperature 
which it represents, three phases — rhombic, monoclinic, and liquid 
sulphur — can coexist in equilibrium. At pressures above this, 
monoclinic sulphur has no stable existence at any temperature 

We have seen that rhombic sulphur may be heated above its 
transition point to a temperature at which it melts, viz. 115°. The 
rhombic is here a metastable phase, and so also is the liquid formed 
by its fusion. We are therefore now dealing with a triple point in a 


metastable region, represented in the diagram by the point B', which 
is the intersection of the prolonged vapour -pressure curves OB and 
DA for rhombic sulphur and liquid sulphur respectively. The dotted 
curve from B' to C represents the effect of pressure on the 
"metastable" melting point of rhombic sulphur, this curve being 
continued in the curve CE for the effect of pressure on the " stable " 
melting point of rhombic sulphur, the possibility of transition into 
monoclinic sulphur ceasing at C. 

The chief features of the diagram may thus be represented as 
follows, metastable conditions being enclosed in brackets : — 

Regions— Di variant Systems. 

BOCE Rhombic 

ECAD Liquid 

BOAD Vapour 

OCA Monoclinic 

Curves — Monovariant Systems. 

BO, (OB') Rhombic, vapour 

OA, (OA') Monoclinic, vapour 

AD, (AB') Liquid, vapour 

OC ..... Rhombic, monoclinic 

AC ..... Liquid, monoclinic 

CE, (CB') Rhombic, liquid 

Triple Points — Nonvariant Systems. 

Rhombic, monoclinic, vapour 

A Monoclinic, liquid, vapour 

C Rhombic, monoclinic, liquid 

(B') (Rhombic, liquid, vapour) 

Monoclinic sulphur offers the peculiarity that its range of exist- 
ence is limited on all sides. It can only exist in the stable condition 
between certain temperatures and certain pressures, the extreme limits 
being given by the temperature and pressure values at O and C. 

Diagrams similar to the sulphur diagram can be drawn for most 
other substances that exist in more than one crystalline modification. 
For example, para-azoxyanisoil yields a similar figure, although one of 
the crystalline modifications is in this case a liquid. The two chief 
triple points are here the points at which the solid crystal passes into 
the liquid crystal, and where the liquid crystal passes into an ordinary 
liquid. The first point is usually spoken of as the " melting point '1 
of the substance, and the second as its transition point. These points 
therefore occur in the reverse order to the corresponding points for 
sulphur, but otherwise the diagram is much the same. 

In the case of a single substance, there is only one point, the triple 
point, at which any three phases can exist together. On this account, 
a system consisting of three phases of a single substance is called a 


nonvariant system, for if we change any of the conditions — here 
temperature or pressure — one or more of the phases will cease to exist. 
When the system consists of two phases, it is said to be monovariant, 
there being for each temperature one pressure, and for each pressure 
one temperature, at which there is equilibrium. When the system 
consists of only one phase, it is said to be divariant, for within 
certain limits both the temperature and the pressure may be changed 
arbitrarily and independently. The regions in the diagram therefore 
correspond to divariant systems ; the curves to monovariant systems ; 
and the triple points to nonvariant systems. 

When the systems considered contain two distinct components, say 
salt and water, and not one, as in the preceding instances, the 
phenomena become more complicated; for here, besides temperature 
and pressure, we have a third condition, viz. concentration, entering 
into the determination of phases. The liquid phase, for example, may 
be pure water, or it may be a salt solution of any concentration up 
to saturation. The phase rule developed by Willard Gibbs furnishes 
us, however, with general methods for treating such systems theoreti- 
cally. It states, for instance, that if the number of phases exceeds 
the number of components by 2, the system is nonvariant. As we 
have seen, this is true for one component, and it is equally true for 
two components. With the components salt and water we have a 
nonvariant system when the four phases, salt, ice, saturated solution, 
and vapour, coexist. There is only one temperature, one pressure, 
and one concentration at which the equilibrium of these four phases 
can take place ; the point at which these particular values are 
assumed is called a quadruple point, and it coincides practically with 
what we have hitherto called the cryohydric point (cp. p. 64). 

Again, the phase rule states that if the number of phases exceeds 
the number of components by 1, the system is monovariant. If, 
therefore, there are three phases with the components salt and water, 
a monovariant system will result. Suppose the phases are salt, 
solution, and water vapour. If we fix one of the conditions, say the 
temperature, the other conditions adjust themselves to certain definite 
values. At the given temperature, the salt solution assumes a definite 
concentration, viz. that of the saturated solution. This solution of 
definite concentration has a definite vapour pressure, less than that 
of pure water. By fixing the temperature, therefore, we also fix the 
concentration and the pressure. Suppose, again, that the three phases 
are ice, salt solution, and water vapour. Let the concentration of the 
solution be fixed, and it will be seen that the temperature and pressure 
adjust themselves to definite values. First, a solution of the given 
concentration can only be in equilibrium with ice at a certain 
temperature fixed by the rule for the lowering of the freezing point 
in salt solutions (cp. p. 63). At this temperature the solution being 
of a fixed concentration will have a vapour pressure defined by the 


law of the lowering of vapour pressure in solutions. By fixing the 
concentration, therefore, we likewise fix the temperature and pressure 
of equilibrium. 

If the number of phases is equal to the number of components, 
the phase rule states that the system is divariant. Let the two 
phases in our example with two components be salt and solution, and 
let the temperature be fixed. The pressure and concentration are 
no longer fixed as in the last case, but may vary in such a way that 
a given variation in the pressure produces a concomitant variation in 
the concentration of the saturated solution. It is necessary, of course, 
that the pressure should be above a certain limiting value in order 
that the third phase, of vapour, should not appear. That the 
concentration of the saturated solution, i.e. the solubility of the salt 
changes with the pressure, has been experimentally ascertained in a 
number of cases. Sometimes the solubility increases with pressure, 
sometimes it diminishes, according as the volume of the solution is 
less or greater than the volume of the solvent and dissolved sub- 
stances separately. If the two phases considered be salt- solution 
and vapour, it is obvious that although the pressure is fixed, the 
concentration and the temperature are not thereby defined. An 
increase of concentration will counteract the effect of a rise in the 
temperature, so that concentration and temperature may be made 
to undergo concomitant variations even though the pressure remains 

If the number of phases is less than the number of components by 
1, the system is then, according to the phase rule, tri variant. In the 
case of two components, the trivariant system has only one phase, and 
with our example of salt and water, the salt solution may be taken as 
the most representative phase, since it contains both components. If the 
temperature and pressure are both fixed, we are still at liberty to vary 
the concentration as we choose, i.e. a change of pressure at the fixed 
temperature causes no concomitant change in the concentration. 
Here, then, we meet with the greatest degree of freedom in varying 
the conditions in the case of two components, as we cannot reduce the 
number of phases further. 

With two components we sometimes get diagrams for melting 
and transition points which closely resemble those obtained for one 
component, when instead of pressure we substitute concentration 
and neglect pressure altogether. Thus with the two components 
paratoluidine and water, we may draw the following diagram 
(Fig. 18). On the vertical axis concentrations are measured instead 
of pressures, and on the horizontal axis temperatures are plotted as 

The line BO represents the concentrations of the solutions in 
equilibrium with solid paratoluidine at different temperatures, i.e. it is 
the solubility curve of solid paratoluidine in water. Under water, 




paratoluidine melts at about 44*2°, slightly lower than the temperature 
of fusion of the dry substance. 1 

If we heat the system above this temperature, the solid phase 
disappears, and a liquid phase takes its place. Now, the liquid phase 
has its own solubility curve LO, and this must cut the solubility 
curve of the solid at the point at which the solid melts. This can be 
shown in the same way as that adopted to prove that water and ice 
have the same vapour pressure at the melting point, by substituting 
in this case solubility for vapour pressure. In general, we may say 
that if two phases are in equilibrium with each other, and one of them 








20° 30° 40° 50 60 70 

Fig. 18. 

is in equilibrium with a third phase, then the second of the original 
pair will also be in equilibrium with the third phase. "We see that 
there is thus considerable resemblance between the melting point of 
a substance under its own saturated vapour and the melting point 
of a substance under its own saturated solution. If we bear in mind 
that concentration of a solution corresponds to pressure of a gas 
(Chap. XVI.), the reason for the resemblance of the solubility and 
pressure diagrams becomes apparent. 

1 The reason for the lower melting point of paratoluidine under water is evident. 
Any substance soluble in water dissolves, when melted under its aqueous solution, a 
portion of the water with which it is in contact. The solid paratoluidine is thus not in 
equilibrium with pure fused paratoluidine, for which the fusing point is highest 
cp. p, 69), but with a solution of water in paratoluidine. 


The similarity is also observable in the case of transition points. 
If we consider the two components, sulphur and an organic liquid 
capable of dissolving it, the general rule referred to in the preceding 
paragraph teaches us that at the transition point of rhombic and 
monoclinic sulphur the solubility of rhombic and monoclinic modifica- 
tions in the solvent must be the same. For if a certain solution is in 
equilibrium with one of the modifications, it must be in equilibrium 
with the second also, since at the transition point the two modifications 
are in equilibrium with each other. We may say shortly, therefore, 
that the vapour- pressure curves and the solubility curves of two 
modifications of the same substance cut at the transition point. This 
actually gives us in some cases a practical method of determining the 
transition point. Sometimes the transition of one modification into 
the other proceeds with such extreme slowness that it is almost 
impossible to observe the transition temperature directly. If, however, 
we investigate carefully the vapour pressures or the solubilities of the 
two modifications at different temperatures, we can construct curves of 
vapour pressure or solubility ; these curves will be found to intersect, 
and the point of intersection may be taken as the point of transition. 

Hydrated salts present many interesting aspects when viewed from 
the standpoint of the phase rule. The components here are the 
anhydrous salt and water, and the number of phases which they form 
may be very great, each solid hydrate being a phase distinct from the 
others. Let us take as our first example sodium sulphate in the form 
of the decahydrate Na 2 S0 4 , 10H 2 O, and the anhydrous salt Na 2 S0 4 . 
The solubility curves of these solids have been already given on p. 51, 
the concentrations being referred to the two components, anhydrous 
salt and water. The two solubility curves intersect at 33°, i.e. the 
two solids are at that temperature in equilibrium with the same 
solution. They must, therefore, according to the rule already given, 
be in equilibrium with each other, i.e. 33° is the temperature of 
transition of the decahydrate phase into the anhydrous phase. This 
may be confirmed directly by heating the decahydrate alone. At 33° it 
melts, but the fusion is not complete, for besides the liquid phase, a 
new solid phase, the anhydrous salt, comes into existence. We have 
therefore at 33° the four phases of decahydrate, anhydrous salt, 
saturated solution, and water vapour, all in equilibrium. This point 
is thus a quadruple point, and as the system consists of two components 
and four phases, it is nonvariant. Consequently, if we alter the 
temperature, pressure of vapour, or the concentration of the solution, 
the equilibrium will be disturbed. If the alteration is only slight and 
temporary, the equilibrium will re-establish itself ; if the alteration is 
permanent, some of the phases will disappear. 

Many instances like the above are known. The essential feature 
is that one hydrate loses water, forming a solution and a lower hydrate 
or anhydrous salt. When this is the case there is a definite transition 




temperature from one hydrate to the other, the higher hydrate, i.e. that 
with the greater amount of water of crystallisation, existing below the 
transition temperature, and the lower hydrate above this point. 

Sometimes a hydrate on being heated melts without separation of 
a new solid phase. An example of this kind is to be found in the 
ordinary yellow hydrate of ferric chloride, Fe 2 Cl 6 , 12H 2 0, already 
referred to on page 68. This hydrate on heating melts completely 
at 37°, the liquid having the same composition as the solid. Here, 
then, we are dealing with a melting point in the ordinary sense, 
the three phases of solid, liquid, and vapour existing together. If the 
system consisted of only one component, the number of phases at 
the melting point would exceed the number of components by 2, and the 
system would be nonvariant. But the number of components is 2, 
and the number of phases at the melting point exceeds the number of 
components by 1 only, so that the system is monovariant according to 
the phase rule. This is as much as to say that we are not fixed down 
to absolutely definite values of temperature, pressure, and concentration 
for the equilibrium of the solid, liquid, and vaporous phases, but may 
alter any one of these conditions within limits, the alteration of one 
being attended by concomitant alterations in the two other factors. 
For example, we may change the concentration of the liquid by adding 
one or other of the com- 
ponents to it. Such a 
change in the composi- 
tion of the liquid phase 
will bring about a 
certain definite change 
in the temperature of 
equilibrium and in the 
vapour pressure. If, on 
the other hand, we alter 
the temperature, it will 
be found that the vapour 
pressure and the concen- 
tration of the liquid 
phase will undergo 
corresponding varia- 

The curves of tem- 
perature and concentra- 
tion for the hydrates of 
ferric chloride are given 
in Fig. 19. In this 
diagram pressure is not 
considered, and the 
curves represent equilibrium curves between solid and liquid phases. 




The line on the left represents the temperatures at which ice is in equi- 
librium with solutions of ferric chloride of various concentrations ; it is, 
in short, the freezing-point curve of ferric chloride solutions (cp. p. 68). 
The curve CAC gives the equilibrium of solid dodecahydrate with ferric 
chloride solution ; it is the solubility curve of the dodecahydrate. The 
point C, where it intersects the ice curve, is the cryohydric point, 
and lies at - 55°. At a temperature of 37° the liquid with which 
the dodecahydrate is in equilibrium has the same composition as the 
dodecahydrate itself. This temperature may therefore be called the 
melting point of the dodecahydrate, and it is the maximum temperature 
at which the hydrate can exist either by itself or in contact with any 
solution of ferric chloride. Addition of either water or ferric chloride 
to the liquid will lower the temperature at which the dodecahydrate 
will separate from the solution. 

If we follow the curve for the dodecahydrate to greater concentra- 
tions, we find that another hydrate may make its appearance at C, 
which lies at about 27°. It will be seen that this point resembles the 
cryohydric point C, inasmuch as it is a quadruple point, the four phases 
being the dodecahydrate, the new heptahydrate Fe 2 G 6 , 7H 2 0, the 
saturated solution, and aqueous vapour. The only difference between 
the equilibrium here and at the cryohydric point is that the two solid 
phases are both hydrates, while at the cryohydric point one of the 
solid phases is ice. The point C corresponds to the point of intersection 
in the sodium sulphate diagram (Fig. 4, p. 51). It is the intersection 
of the solubility curves of the dodecahydrate and the heptahydrate, 
and therefore represents the transition point of these two phases. An 
investigation of the curve of the heptahydrate shows that it is of the 
same nature as the curve of the dodecahydrate. It reaches a maximum 
as before, the concentration of the solution and the temperature there 
being equal to the composition and melting point of the hydrate. This 
curve for the heptahydrate finally cuts the curve of a lower hydrate, 
and similar curves are repeated until at last the solubility curve of the 
anhydrous salt is reached. The curve for each hydrate reaches a 
maximum temperature, which is the melting point of the hydrate, 
and cuts the curves for other hydrates at temperatures which are 
transition temperatures. 

In the preceding instances we have seen how water may be 
removed from hydrates by continually raising the temperature, solu- 
tion being at the same time present. Now, it is possible in many 
cases to remove the water of crystallisation from a hydrate without 
any solution being formed at all. This can be done most conveniently 
by placing the hydrate in an evacuated desiccator over a substance 
such as sulphuric acid or phosphorus pentoxide. The hydrate is at a 
given temperature in equilibrium with a small, and in most cases 
measurable, pressure of water vapour, i.e. it has a vapour pressure just 
as a solution has. If the pressure of water vapour above the hydrate 





SO mm. 

is kept beneath this value, the hydrate will lose water and be con- 
verted into a lower hydrate or the anhydrous salt. In an evacuated 
desiccator containing phosphorus pentoxide the pressure of water 
vapour is practically kept at zero, so that the loss of water by the 
hydrate goes on continuously. Copper sulphate in the form of the 
pentahydrate CuS0 4 , 5H 2 0, for example, gradually loses water under 
these conditions, and is converted into the greenish-white monohydrate 
CuS0 4 , H 2 0. The vapour pressure of this hydrate is so small at the 
ordinary temperature that it remains practically unchanged in the 

In this mode of dehydration of a hydrate, we have only three 
phases coexisting, viz. the higher hydrate, the lower hydrate, and 
aqueous vapour. The liquid 
phase is entirely wanting. 
Now, the system consists of 
two components, the anhy- 
drous salt and water, so 
that the number of phases 
exceeds the number of com- 
ponents only by one. The 
system is therefore mono- 
variant, i.e. we can change 
one of the conditions with- 
out destroying the equili- 
brium altogether, the other 
conditions at the same time 
undergoing concomitant 
alterations. It should be 
noted that the condition of 
concentration is here practi- 
cally absent, for there is no 
phase present in which the 
concentration varies continuously as it does in a solution. The effect of 
passing from one hydrate to another at constant temperature is seen in 
the accompanying diagram (Fig. 20), which represents the dehydration 
of copper sulphate pentahydrate at 50°. The dehydration does not 
proceed in one step from the pentahydrate to the anhydrous salt, but 
in three stages, two intermediate hydrates being formed. Each of 
these hydrates has its own vapour pressure ; and where two hydrates 
coexist, the observed vapour pressure is the vapour pressure of the 
higher hydrate. Pressures have been tabulated on the vertical axis, 
and composition in molecules on the horizontal axis. Until the 
molecule of copper sulphate has lost two molecules of water, the vapour 
pressure remains constant at 47 mm., after which there is a sudden 
drop to 30 mm. The first of these values is the pressure of the 
pentahydrate ; the second is the pressure of the trihydrate formed as 


6 H 2 

3 H 2 1 H 2 


0H 2 


the first step in the dehydration. This value of the pressure is 
retained until two more molecules of water have been lost, when it 
sinks suddenly to 4 '5 mm. This indicates that a monohydrate, with 
a vapour pressure of 4*5 mm. has been formed. Further dehydration 
produces no diminution of the pressure until all the water has been 
lost, when, of course, the pressure altogether disappears. This method 
of systematic measurement of vapour pressure during the dehydration 
of a hydrate at constant temperature can be used to ascertain the 
existence of intermediate hydrates, which may not be easily prepared 
in other ways. 


Fig. 21. 

If tlje dehydration were conducted at another temperature than 
50°, a similar diagram would result ; the values of the pressures for the 
different temperatures would, however, be all higher or all lower than 
before. As the system with three phases is a mono variant one, the 
temperature and the pressure may be altered without the equilibrium 
being destroyed, but to a given alteration of the one there corre- 
sponds a definite alteration of the other. Each hydrate, therefore, has 
a vapour-pressure curve precisely like that of liquid water. These 
curves are represented in the temperature-pressure diagram of Fig. 21 
by the lines OM, OT, OP, for the monohydrate, trihydrate, and 
pentahydrate respectively. The vapour-pressure curve of ice is repre- 
sented by the line OA, and that of water by the line AW, the point 
A where these curves intersect being the freezing point, or more 


correctly the triple point. SC is the curve of vapour pressures of 
solutions saturated with the pentahydrate at different temperatures. 
This curve has a smaller vapour pressure than that of pure water, 
and consequently cuts the curve for ice at a temperature below 
the freezing point. The line SC is the curve for the equilibrium 
of the three phases, pentahydrate, solution, and vapour. At the 
point C, where it cuts the ice curve, the three phases are also in 
equilibrium with ice, so that C represents the cryohydric point for 
copper sulphate. As the lower hydrates cannot exist in contact with 
ice or an aqueous solution in stable equilibrium, these curves do not 
cut the lines CS or OC at all, unless, indeed, we represent them as all 
meeting OC at 0, the point at which the pressure becomes zero, as 
has been indicated in the diagram. 

If the pressure of water vapour does not reach the vapour pressure 
of the monohydrate, copper sulphate will exist as the anhydrous salt. 
It can exist then as anhydrous copper sulphate in contact with water 
vapour at any point in the region beneath MO. If the pressure of water 
vapour is equal to the vapour pressure of the monohydrate, this salt 
can exist in presence of the anhydrous salt and water vapour at points 
on the curve MO. If the pressure is greater than the vapour pressure 
of the monohydrate, the anhydrous salt ceases to exist, and passes into 
the monohydrate. The region of existence of the monohydrate is 
MOT, the lines MO and TO bounding this region indicating the 
pressures at which it can coexist with the anhydrous salt and the next 
higher hydrate respectively. Similarly, TOP is the region of the tri- 
hydrate, OP giving the pressures at which it can coexist with the 
pentahydrate. The region of this, the highest, hydrate is POCS. 
The form of this region is different from that of the previous regions, 
because the pentahydrate phase can at certain pressures and tempera- 
tures coexist with ice as is represented by the line OC. If the pressure 
of water vapour is increased to values above those given by the vapour- 
pressure curve of the pentahydrate CS, some of the vapour will con- 
dense with formation of a new phase, viz. solution. The region of the 
existence of solutions is SCAW, bounded by the vapour-pressure curve 
of the saturated solutions, of ice, and of pure liquid water respectively. 
The diagram throws some light on the behaviour of hydrated salts 
when exposed to an atmosphere containing the ordinary amount of 
moisture. The pressure of water vapour in a well-ventilated labora- 
tory in this country is about 8 to 10 mm. on the average. If the 
vapour pressure of a hydrate is greater than this amount at the 
atmospheric temperature, the hydrate will lose water, i.e. will 
effloresce. This is the case, for example, with common washing soda, 
]STa 2 C0 3 , 10H 2 O, which, when exposed to the atmosphere, loses water 
in the form of vapour, with production of a lower hydrate. If, on 
the other hand, the pressure of water vapour in the atmosphere is 
greater than the vapour pressure of the hydrate, the water vapour 



may condense, and a higher hydrate or a solution may be formed. 
Thus, if anhydrous copper sulphate, or one of the lower hydrates of 
this salt be exposed to the atmosphere, the water vapour will be 
slowly absorbed with ultimate formation of the pentahydrate, for all 
the hydrates of copper sulphate have a lower vapour pressure than 
the pressure of the water vapour usually found in the atmosphere. If 
calcium chloride or its common hydrate, CaCl 2 , 6H 2 0, is exposed to 
the air, it deliquesces, i.e. forms a liquid phase. Here the vapour 
pressures of the hydrate and of the saturated solution are lower than 
the pressure of water vapour commonly in the atmosphere, amounting 
to only 2 or 3 mm. at the ordinary temperature. The result is that a 
solution is formed which will become more and more dilute by absorp- 
tion of water vapour, the process coming to an end when the vapour 
pressure of the solution is equal to the pressure of water vapour in 
the atmosphere. 

Formation of New Phases. — Under conditions where a new phase 
may appear, it does not necessarily follow that it must appear. When a 
crystalline solid is heated to its melting point, it invariably melts if any 
further heating is attempted. Here we have the new phase, the liquid, 
making its appearance as soon as the conditions are such that its stable 
existence becomes possible. If we cool the liquid, on the other hand, 
we may easily reach temperatures considerably below its freezing point 
without any solidification actually taking place. Here the new phase, 
the crystalline solid, does not appear when its existence becomes possible, 
but may remain unformed for an indefinite period. We meet with the 
same reluctance to form new phases at transition points. Rhombic 
sulphur can exist at temperatures above 95*6°, the transition point 
into monoclinic sulphur, and the latter may remain for a long time 
unchanged even at the ordinary temperature, which is far below the 
transition point into the rhombic modification. 

New hydrates of well-known substances are constantly being dis- 
covered since investigations have been directed to their formation, 
although it is practically certain that conditions compatible with their 
existence must have previously been encountered in actual work with 
these substances. 

Although sodium sulphate in the form of the decahydrate is efflor- 
escent under ordinary atmospheric conditions, its vapour pressure being 
greater than the pressure of water vapour in the atmosphere, yet it 
may, if perfectly pure, remain for a long time in the air without a trace 
of efflorescence being observable. On the other hand, there are many 
salts which are under the atmospheric conditions capable of taking up 
moisture to form a higher hydrate, or even a solution, and yet remain 
quite unaffected in the air. In each case, removal or absorption of 
water would result in the formation of a new phase, but the tendency 
to the formation of the new phases is so small that their formation 
may be delayed or never occur at all. It must be borne in mind that 


the reluctance to the production of new phases only applies to the 
-first appearance of the new phase. As soon as the smallest particle of 
lit appears, or is introduced from without, its formation goes on steadily, 
-and in many cases very rapidly. Supersaturated solutions of sodium 
thiosulphate, for example, may be kept for years without showing any 
^tendency to crystallise, but if the merest trace of the solid crystalline 
phase is introduced, the whole mass becomes solid in the course of a 
few seconds. Similarly water, if perfectly air-free, may be heated to 
a temperature much above its boiling point, but in such a case, when 
■the smallest bubble of vapour is formed in the interior of the liquid, 
] the whole passes into the new vaporous phase with explosive violence. 
When ice and a salt (or other substance soluble in water) are 
brought together at a temperature below the freezing, point, there is 
•the possibility of the formation of a new phase — the solution — and this 
i phase generally forms, the temperature then under favourable condi- 
tions falling to the cryohydric point. It is questionable, however, if 
this is invariably the case ; and it seems quite possible that two sub- 
stances in the solid state might be brought together at a temperature 
ibove the cryohydric point without liquefaction taking place. 

It is evident from what has been said in this chapter that it is 
lot always the phase most stable under the given conditions which 
ictually exists. A metastable phase may exist for an indefinite time 
without passing into the most stable phase, provided that this latter 
phase is entirely absent. As soon as the stable and metastable phases 
ire brought into contact, however, the former begins to be produced at 
;he expense of the latter. The transition from the metastable to the 
itable phase takes place as a rule fairly rapidly, but in some instances 
fthe transformation is so slow as to be practically unobservable. 
strongly overcooled liquids, for example, crystallise with extreme 
lowness, even after they have been brought together with the stable 
Tystalline phase (cp. p. 62). The crystalline modifications of silica 
quartz and tridymite), are at ordinary temperatures so far beneath 
I heir temperature of transformation, if such exists, that they show no 
endency to reciprocal transformation, and both forms must be accounted 
table. The same holds good for the two forms of calcium carbonate, 
.rragonite and calcspar. Yellow phosphorus is at ordinary tempera- 
ures metastable with regard to red phosphorus, which is the stable 
orm, yet in the dark it keeps for an indefinite time without under- 
going much alteration, even although it may be in contact with red 

When we come to inquire what phase will be formed when there 
5 the possibility of formation of several different phases, we find that 
'> is not, as we might be inclined to expect, always the most stable 
ase that is formed, but rather a metastable phase, which may there- 
ter pass into the stable phase. A substance, then, in passing from 
n unstable to the most stable phase very frequently goes through 


phases of intermediate degrees of stability, so that the transformation 
does not occur directly, but in a series of steps. Liquid phosphorus, 
for instance, which is itself metastable with regard to red phosphorus, 
does not on cooling pass into the latter, most stable, modification, but 
into the metastable yellow phosphorus. Molten sulphur, again, when 
quickly cooled by pouring into cold water, does not pass directly 
into the stable rhombic sulphur, but into the comparatively unstable 
plastic sulphur, which then in its turn undergoes transformation into 
more stable varieties. It is a common experience in organic chemistry 
to obtain substances first in the form of oils which afterwards 
crystallise, sometimes only after long standing. The alkali salts of 
organic acids, for example, on acidification with a mineral acid in 
aqueous solution, very frequently do not yield the free acid in the 
most stable solid form, but as an oily liquid, which crystallises with 
more or less rapidity. Thus if solid paranitrophenol is dissolved in 
caustic soda solution, and the solution then acidified with hydrochloric 
acid, the paranitrophenol is liberated as an oil which crystallises after 
a few minutes. 

If we consider that the least stable phase of a substance has always 
the greatest vapour pressure, or, if we are dealing with solutions, the 
greatest solubility, the formation of intermediate metastable phases is 
not perhaps so unaccountable as at first sight appears. In the above 
instance of paranitrophenol, the system before acidification consists of ' 
a liquid phase only, since for our present purpose we may neglect the j 
vapour phase altogether. On acidification, the new phase may not 
make its appearance for some moments, owing to the general reluctance 
exhibited in the formation of new phases, and the new phase which , 
eventually does make its appearance is that which entails least altera- 1 
tion in the system, i.e. that which leaves most in the solution. In i 
other words, the most soluble and least stable phase is formed first, 
the less soluble and most stable phase only appearing as a product of 
the transformation of the former. 

A very complete, non-mathematical, exposition of the subject dealt with 
in this chapter is given by W. D. Bancroft in his book The Phase Rule 
(Ithaca, New York). 



A CHEMICAL change is almost invariably attended by a heat change, the 
latter being generally of such a nature that heat is given out during the 
progress of the action. Vigorous reactions are accompanied by con- 
siderable evolution of heat ; in feeble reactions, on the other hand, the 
heat evolution is comparatively small as a rule, and in some cases gives 
place to heat absorption. In special circumstances there might be 
' neither evolution nor absorption of heat, but instances of this are rare 
or altogether wanting. 

From the fact that vigour of chemical action frequently goes 
hand in hand with heat evolution, it was at one time thought that 
measuring the amount of heat evolved in any given action was tanta- 
mount to measuring the chemical affinity of the substances taking part 
in the action ; but this point of view has of late years been entirely 
given up owing to practical difficulties in reconciling it with the facts, 
and to a general advance in our theoretical knowledge of the subject. 
If heat evolution were to be taken as an accurate measure of chemical 
affinity, there is an obvious difficulty in explaining why certain 
changes take place with absorption of heat, since this would correspond 
J to a negative chemical affinity, and there would therefore be no reason, 
chemically speaking, why the action should take place at all. By 
introducing the heats of attendant physical changes in a somewhat 
arbitrary way, it was found possible to explain away the exceptions, 
but the explanations were in many instances so laboured that it became 
expedient to drop the rule altogether, in the strict sense, and be con- 
tent with the recognition of a general parallelism between the amount 
of heat evolved in an action and the readiness with which it takes 

The amount of heat change attendant on a chemical change is 
perfectly definite in ordinary circumstances, and is easily susceptible of 
exact measurement. A gram of zinc when dissolved in sulphuric acid 
will always occasion the same heat development if the conditions of 
the chemical action are the same. If the conditions are different, the 



thermal effect will also be different. Thus it is necessary in the first 
place to ensure that in each case exactly the same chemical action 
occurs. The action of zinc on sulphuric acid differs according as 
the acid is concentrated or dilute. In the former case, zinc sulphate 
and sulphur dioxide are the chief products, in the latter case zinc 
sulphate and hydrogen. These are essentially different chemical 
actions, and evolve different amounts of heat for a given quantity of 
zinc dissolved. But even if we ensure that the only products are zinc 
sulphate and hydrogen, there will still be a difference in the heat 
development if the sulphuric acid in two cases is at different degrees of 
dilution. The difference here, however, will be slight, and may for 
most purposes be neglected. Again, a difference in the temperature 
at which the action takes place will occasion a difference in the heat 
evolution ; but in this case also the difference is comparatively slight,, 
and negligible for small variations of temperature. Lastly, if the zinc 
and sulphuric acid form part of a voltaic circuit, as in a Daniell or a 
Grove cell, the heat evolution is then very different from what it is if 
the chemical action is not accompanied by the generation of an electric 

From the standpoint of the conservation of energy, these phenomena 
are easily understood. Each substance, under given conditions, pos- 
sesses a certain definite amount of intrinsic energy, so that if we 
are dealing with a system of substances, a definite amount of energy 
is associated with that system as long as it remains unchanged. If it 
changes now into another group of substances, each of these will have 
its own intrinsic energy, and the new system will in general have a 
different amount of energy from that of the original system. Suppose the ' 
second system has less energy than the first. From the law of con- 
servation, it is plain that the difference of energy between the two 
systems cannot be lost, but must be transformed into some other kind 
of energy. Now the energy difference between two systems is usually 
accounted for as heat, and in our example the heat evolved during the 
solution of zinc in dilute sulphuric acid measures the difference of the I 
intrinsic energy of the zinc and dilute sulphuric on the one hand, and 
hydrogen and dilute zinc sulphate on the other. If pure sulphuric is 
taken instead of a mixture of sulphuric acid and water, the second 
system is now sulphur dioxide and zinc sulphate, mostly in the solid 
anhydrous state — a system which has quite a different amount of j 
intrinsic energy associated with it from hydrogen and dilute aqueous 
solution of zinc sulphate, so that the energy differences (and therefore 
the heat evolved) are widely divergent in the two cases. When the 
zinc and sulphuric acid form part of a galvanic cell, the initial and 
final systems are the same as above, so that there is the same energy 
difference as before ; but now all the energy does not pass into heat, 
some of it being transformed into electric energy, which takes the shape 
of an electric current passing outside the system. The consequence 



is that much less heat is obtained by the solution of the zinc in this 
case than was obtained when no electric current was generated. 

Dealing now with smaller heat effects, we find that the intrinsic 
energy of a system is not the same at one temperature as it is at 
another; for if we wish to raise the temperature we must supply 
energy in the form of heat to the system, the quantity supplied depend- 
ing on the heat capacity of the substances which compose the system. 
In changing from one system to another, therefore, at different tempera- 
tures, different amounts of heat will be evolved, for in general the 
heat capacities of the two systems will be different. If we dilute a 
solution of sulphuric acid or of zinc sulphate, we find that a heat 
change accompanies the process, and as this heat of dilution is not as 
a rule the same for two substances, the total thermal effect depends on 
the concentration of the solutions employed. 

From the principle of the conservation of energy we see that if we 
have in a chemical change the same initial system and the same final 
system, the same thermal effect will always be produced no matter 
how we pass from the first system to the second, provided that no 
other form of energy than heat is concerned in the transformation. 
Hess, who originally worked this out experimentally, gives the follow- 
ing numerical example. Pure sulphuric acid was in one experiment 
neutralised with ammonia in dilute aqueous solution ; in other experi- 
ments it was first of all diluted with varying amounts of water before 
neutralisation, the heats of dilution and the heats of neutralisation 
being noted in each case. The experiment resulted as follows : — 

Mols. Water 

Heat of Dilution. 

Heat of Neutralisation. 
















The first column gives the number of molecules of water which were 
added to one molecule of sulphuric acid ; the second gives the number 
of heat units evolved on the addition of the water ; and the third gives 
the number of heat units evolved on the neutralisation of the resulting 
solution by dilute ammonia. It will be noticed that the sum of the 
two heats is very nearly the same in the four cases, for in each the 
starting-point is from pure sulphuric acid and dilute ammonia, and 
the product is dilute ammonium sulphate. 

This constancy of the total heat evolved is frequently made use of 
in thermochemistry for the determination of heat changes not easily 
accessible to direct measurement. Yellow phosphorus, for example, 
on conversion into red phosphorus is known to give out a considerable 
amount of heat, but the direct determination of this amount is a matter 
of some difficulty. An indirect determination, on the other hand, may 
be made with the greatest ease. Favre found that when a gram atom 
of yellow phosphorus is oxidised to an aqueous solution of phosphoric 


acid by means of hypochlorous acid, the oxidation is attended by the 
disengagement of 2386 heat units. A gram atom of red phosphorus 
in similar circumstances yields 2113 heat units. If now a gram atom 
of yellow phosphorus were first converted into red phosphorus, and 
this then oxidised to phosphoric acid, the total heat evolution would 
be 2386 heat units, since the sum must be equal to the heat evolved 
in the direct oxidation. But the second part of the action, viz. the 
oxidation of the red phosphorus, yields 2113 units, so the first stage 
of the action, viz. the transformation of the yellow into the red 
phosphorus, must yield 2386-2113 = 273 units. 

Of the heat units referred to on p. 6, the most convenient for 
thermochemical purposes is the centuple unit, denoted by K, which is 
the quantity of heat required to raise the temperature of 1 gram of 
water from 0° C. to 100° C. With this unit, which has the advantage 
of being easily determined practically, the ordinary heats of reaction 
are represented by numbers such as those in the preceding paragraph, 
not inconveniently large and not requiring the use of fractional values, 
since the degree of accuracy in the experimental determinations corre- 
sponds to about one unit. 

We have no means of determining the amount of intrinsic energy 
in any substance ; we can only measure differences between the 
intrinsic energies of certain substances, or systems of substances. If 
all substances were mutually convertible, directly or indirectly, we 
might take one substance as standard, and refer all intrinsic energies 
to it by means of numbers stating the quantity of energy possessed 
by the substance in excess of the standard. But chemical substances 
are not mutually convertible without restriction. In particular, the 
elements cannot be converted into each other by any means in our 
power. We are therefore unable to compare the intrinsic energies of 
the elements together, and so for purposes of calculation we may adopt 
any value for them that we please. The easiest system is to make the 
intrinsic energies of all the elements equal to 0, and refer all other 
intrinsic energies to this value for the elements. If in the equation 

Pb + I 2 = PbI 2 , 

we take the ordinary chemical symbols of the elements and compounds 
as signifying the amounts of intrinsic energy in the substances, as well 
as the quantities of the substances themselves, the equation does not 
balance, for in the conversion of lead and iodine into lead iodide there 
is heat evolution, viz. 398 K for one gram atom of lead, so that the 
equation to be an accurate energy equation should read 

Pb + I 2 = PbI 2 + 398 K. 

The intrinsic energy of a gram molecule of lead iodide is 398 K less than 
the sum of the intrinsic energies of the atoms from which it is formed, 
and is therefore equal to - 398 K, since the sum of the intrinsic energies 



of the elements is zero. If we actually write the amounts of energy 
associated with the various substances, we have the equation 

Pb + I 2 = Pbl 2 
+ = - 398 K + 398 K. 

Now, the heat given out on the production of lead iodide, or any other 
substance from its elements, is called the heat of formation of the 
substance, and we see from the above instance that this must be equal 
to the intrinsic energy of the substance with the sign reversed \ for on 
the left-hand side of the equation the sum of the energies is always 
equal to zero, being the intrinsic energy of elements alone, so that the 
sum of the energies on the left-hand side must also be zero, and the 
intrinsic energy of the compound thus equal to its heat of formation 
with the sign reversed. The heats of formation of compounds from 
their elements are for this reason very important in thermochemical 
calculations, their practical use being as follows. If from the sum of 
the heats of formation on the right hand of an ordinary chemical 
equation we subtract the sum of the heats of formation on the left 
hand, we obtain the heat given out or absorbed during the reaction. 
If the difference has the positive sign, the heat is evolved ; if it has the 
negative sign, the heat is absorbed. If we reverse the signs of the 
heats of formation, i.e. if we write the values of the intrinsic energies, 
and subtract the sum on the right hand from the sum on the left, 
we arrive at the same result. 

As an example, we may take the displacement of copper from copper 
sulphate by metallic iron according to the equation 

Fe + CuS0 4 , Aq = Cu + FeS0 4 , Aq • 

where Aq indicates that the substance to whose formula it is attached 
is in aqueous solution. The heat of formation of copper sulphate in 
solution is 1984 K, and of ferrous sulphate under the same conditions 
2356 K. The two metals have of course no heats of formation. If 
we subtract, therefore, the heat of copper sulphate from that of ferrous 
sulphate, we get the heat of reaction required, viz. 372 K. Writing 
the equation with the values of the intrinsic energies, we have 

Fe + CuS0 4 = Cu + FeS0 4 

- 1984 K = - 2356 K + 372 K. 

If we know the heat of a reaction and the heats of formation of all 
the substances but one concerned in the action, we can calculate the 
heat of formation of that substance directly from the energy equation. 
For example, the heat of neutralisation of hydrochloric acid by caustic 
soda, when both substances are in aqueous solution, is 137 K, the heats 
of formation of dissolved hydrochloric acid, dissolved caustic soda, and 
liquid water respectively being 393 K, 1118 K, and 683 K. If we let 


x represent the unknown heat of formation of sodium chloride in 
aqueous solution, we obtain the following equation : — 

HC1, Aq + NaOH, Aq = NaCl, Aq + H 2 
-393K- 1118 K = -x - 683K + 137K, 

whence x= 965 K. 

When an element like sulphur exists in more than one modification 
it is necessary to specify which modification we assume to have zero 
intrinsic energy, as there is always a heat change in passing from one 
modification to another. As a rule, the commonest or most stable 
variety is taken as the standard as convenience dictates. Heats of 
formation of sulphur compounds are generally referred to rhombic 
sulphur ; those of phosphorus compounds to yellow phosphorus. 

In the case of carbon compounds we seldom deal directly with heats 
of formation, but rather with heats of combustion, on account of 
their practical importance, and also on account of the ease with which 
they can be determined. The heat of formation, however, can easily 
be calculated from the heat of combustion. We find, for example, that 
methane has a heat of combustion equal to 2138 K, the products of 
combustion being carbon dioxide and water. Now, the heat of forma- 
tion of carbon dioxide from carbon in the form of diamond is 943 K, 
and of water 683 K. For the heat of formation of methane we have 
therefore the following equation : — 

CH 4 + 20 2 = C0 2 + 2H 2 

- x + = - 943 K - 2 x 683 K + 2138 K. 

whence x = 171 K. We see from this that the combustion of methane 
gives out less heat than we should get by burning the same quantity 
of carbon and hydrogen as the free elements, and this is true of most 
carbon compounds. The difference between the heat of combustion of a 
hydrocarbon and that of the carbon and hydrogen composing it is not 
as a rule very great, so that a calculation of the latter gives an approxi- 
mate value for the former. Thus the heat of combustion of the carbon 
and hydrogen in amylene, C 5 H 10 , would be 5 x 943 K+5x 683K = 
8130 K. The heat of combustion of amylene vapour was found- 
by direct experiment to be 8076 K, a number differing only slightly 
from the preceding one. 

When a carbon compound contains oxygen as well as hydrogen, 
its heat of combustion may be roughly determined by means of 
"Welter's rule." According to this rule, the oxygen is subtracted 
from the molecular formula together with as much hydrogen as will 
suffice to convert it completely into water, the heat of combustion of 
the carbon and hydrogen in the residue then giving an approximate 
value of the heat of combustion of the whole compound. As an 
example we may take propionic acid, C 3 H 6 2 , whose heat of combustion 
has been found by direct experiment to be 3865 K. If we subtract- 


2H 2 from the molecular formula, we are left with the residue C 3 H 2 , 
the elements of which have the heat of combustion 

3 x 943 + 683 K = 3502 K. 

It is evident that the approximation is here by no means close, the 
error in this case being about 10 per cent. A better result may 
usually be obtained by subtracting the oxygen, not with the cor- 
responding quantity of hydrogen, but with the corresponding 
quantity of carbon, and then estimating the heat of combustion of 
the elements in the residue. In the above example we subtract C0 2 
from the formula C 3 H 6 2 , and have C 2 H 6 left as residue. This gives 
the heat of combustion 2x943 + 3x683 = 3935K, a much better 
approximation to the experimental value. As another instance of the 
two methods of calculation, we may take cane sugar, C ]2 H 22 O n . By 
Welter's rule we subtract 11H 2 from the molecule, and for the 
residue C 12 get the heat of combustion 12x943 = 11, 316 K. By the 
other method we subtract 5"5C0 2 in order to dispose of the 11 atoms 
of oxygen, and obtain the residue 6*5C and 11H 2 . The heat of 
combustion of these quantities of the elements is 6*5 x 943 + 11 x 683 = 
13,642 K. The value experimentally found is 13,540 K, a number 
much closer to the second calculated value than to the first. 

Some hydrocarbons have a greater heat of combustion than that of 
the carbon and hydrogen contained in them. The heat of combustion 
of acetylene, for example, is 3100 K; the heat of combustion of the 
two atoms of carbon and the two atoms of hydrogen contained in its 
molecule being 2x943 + 683 = 2569K. This corresponds to a heat 
of formation of -531 K, i.e. this amount of heat is absorbed on 
formation of acetylene from its elements. We have here, then, an 
example of an endothermic compound formed from its elements with 
heat absorption, in contradistinction to the bulk of compounds, which 
are exothermic, i.e. are formed from their elements with evolution of 
heat. Other common examples of endothermic compounds are carbon 
disulphide, which is formed with a heat absorption of 287 K, and 
gaseous hydriodic acid, which is formed with a heat absorption of 6 1 K. 
Endothermic compounds like these are comparatively unstable, and 
give out heat on their decomposition. Hydriodic acid gas, for instance, 
is decomposed by gentle heating ; carbon disulphide can be split up 
into its elements by mechanical shock ; and carbon and hydrogen may 
be regenerated from acetylene by the passage of electric sparks through 
the gas. The substance hydrazoic acid, or azoimide, N 3 H, has a large 
negative heat of formation, which is no doubt closely associated with 
its extremely explosive properties. 

Such endothermic compounds are formed directly from their 
elements with difficulty, if they can be formed at all. At ordinary 
temperatures their direct formation does not take place, but if the 
elements are brought into contact at a very high temperature, then 


combination may occur. Thus carbon disulphide is formed by passing 
sulphur vapour over red-hot carbon. Acetylene is produced when 
carbon and hydrogen are brought into contact at the very high 
temperature of the electric arc. This behaviour is exactly opposite 
to what we find with the common exothermic compounds, which are 
stable enough at ordinary temperatures, but are frequently decomposed 
by high temperatures. 

It has already been stated that for thermochemical purposes it is 
necessary to specify exactly the condition of each of the substances 
concerned in the action under discussion. This is not only true for 
the chemical condition, but also for the physical state of the substances. 
We must know whether the substances are in the solid, liquid, or 
gaseous states, or, if they are in the state of solution, in what solvent, 
and at what dilution. This is so because change of physical state is 
accompanied by heat change, which must be taken into account in 
thermochemical investigations. Liquid sulphur, on combining with 
oxygen to give sulphur dioxide, will not give out the same amount 
of heat as rhombic sulphur, for the latter on melting absorbs about 
3 K, which must therefore be added to the heat of combustion of the 
rhombic sulphur. The correction for the difference between the solid 
and liquid states is often small, and never amounts to more than about 
50 K. If the substance is in the state of vapour, the heat of vaporisa- 
tion must be added to the thermochemical data for the liquid. This 
correction is often considerable, amounting approximately to one-fourth 
of the value of the boiling point of the substance on the absolute 
scale (Trouton's rule). Thus the correction for water according to this 
rule would be 0*25 x 373 = 93 K, the actual heat of vaporisation at 100° 
being 97 K. The heat of formation of liquid water at the ordinary tem- 
perature from oxygen and hydrogen is 683 K, the number we have used 
throughout in the above calculations. At 100° the heat of formation of 
liquid water is somewhat less, viz. 676 K. If, now, we want to find the 
heat of formation of gaseous water at 100°, we must subtract the heat 
of vaporisation of the liquid, viz. 97 K, and thus obtain 579 K as the 
heat of formation of water vapour. There is still another circumstance 
which must be taken into consideration when a chemical action is 
accompanied by the disappearance or formation of gases ; or, in 
general, when the action is accompanied by a great change of volume. 
Each gram molecule of gas generated performs an amount of work 
equal to 0*02r K, for, as we have seen, the equation pv = RT becomes 
pv-2T for the gram molecule, and B has the value 2 in small calories, 
or 0*02 in centuple calories (see p. 29). This amount of heat, then, 
is absorbed on production of the gas. If, on the other hand, a gram 
molecule of gas disappears, a corresponding amount of heat is pro- 
duced in the action. At 27° the actual amount per gram molecule is 
0*02 x (27 + 273) = 6 K, the value being the same for all gases. This 
correction is of importance in the case of carbon compounds, which, 


under ordinary circumstances, are burned at the atmospheric pressure, 
the volume increasing considerably during the combustion. The actual 
thermochemical measurement is usually made, on the other hand, in 
a closed " calorimetric bomb," the volume thus remaining constant. 
If we consider the combustion of benzene, for instance, we have the 
following volume relations : — 

C 6 H 6 + 90 2 =6C0 2 +3H 2 0; 

9 vols. 6 volsT 

or, if the substances are all in the gaseous state, 
C 6 H 6 + 90 2 =6C0 2 +3H 2 

1 vol. 9 vols. 6 vols. 3 vols. 

Each volume in the above equations is the gram-molecular volume, the 
volume of the liquid substances being negligible. If both the benzene 
and the water formed by its combustion are in the liquid state, there 
is a shrinkage of three volumes on completion of the combustion. If 
all the substances are in the gaseous state, there is a shrinkage of only 
one volume. On the supposition that the combustion takes place in 
the calorimetric bomb at 27°, with evolution of m centuple calories, then 
if the liquid benzene is burned at constant pressure, we shall have a 
heat evolution of m+ 18K. Suppose now that the benzene is burned 
as vapour at 27° by passing a stream of air or oxygen through the 
liquid and igniting the mixture at a jet, the water vapour being carried 
off without condensing ; and suppose further that the heat of vaporisa- 
tion of benzene and water at this temperature are b and w respectively, 
then we can calculate the heat of combustion under these conditions 
as follows. To vaporise the gram molecule of benzene, b heat units 
are absorbed, and this heat is given out again when the benzene ceases 
to exist as such, so to the heat of combustion of the liquid we must 
add this heat of vaporisation. But the water obtained in the previous 
case was liquid water, in the formation of which from vapour there was 
evolved w heat units per gram molecule. This amount of heat is not 
given out if the water remains in the gaseous state, so that from the 
heat of combustion given above we must now subtract 3w. Finally, 
there is now a shrinkage of only one volume, so that there is to be 
added 6 K to the heat of combustion at constant volume. The heat 
of combustion under the circumstances is therefore m + b - 3w + 6K. 

The apparatus used to measure heats of chemical change is 
essentially the same as that used in physics for measuring heat 
quantities, and in particular the water calorimeter is universally 
employed. The chemical action is allowed to take place in a chamber 
immersed in a known amount of water of known temperature, and the 
change of temperature brought about in this water by the chemical 
action is noted. As the apparatus itself, viz. vessels, thermometers, 
stirrers, etc., is heated along with the water it contains, its water 
equivalent, i.e. the quantity of water which has the same heat capacity 


as the apparatus, must be determined and added to the quantity of 
water actually employed in the experiment. This can be done by 
adding a known quantity of heat to the apparatus and ascertaining 
the resultant change of temperature in the water of the calorimeter. 

The chief source of error in such experiments lies in heat exchange 
with external objects by conduction and radiation. To reduce this 
error to a minimum, the chemical action must be made to go as fast 
as possible, and the temperature of the calorimeter must never be 
allowed to depart greatly from the temperature of the room in which 
the experiment is made. Conduction is avoided by having the 
calorimeter surrounded by one or two vessels with stagnant air spaces 
between them, contact between the vessels being made by a bad heat 
conductor, such as cork, and reduced to as few points as possible. 

If the calorimeter is constructed to contain half a litre of water, 
the heat capacity of the apparatus is small in comparison, and by the 
use of a thermometer which can measure differences of temperature to 
a thousandth of a degree, very accurate results can be obtained with 
a relatively small expenditure of material. The most convenient form 
for the calorimeter is that of a cylinder, whose height is one and a 
half times to twice its diameter, so that in many cases an ordinary 
beaker serves the purpose very well, a larger beaker with a cover being 
the surrounding vessel. 

For further information concerning the methods and results of thermo- 
chemistry, the student may consult Muir and Wilson, Elements of Thermal 



I In the homologous series of organic chemistry, for example the series 
of the saturated alcohols, there is a close resemblance in chemical 
properties amongst the members, so that it is possible to give general 
methods for the preparation of the substances and general types of 
action into which they enter. The actual readiness with which the 
substances are formed or are acted on by other substances, may, and 
usually does, differ from case to case, there being a gradation in 
chemical activity as successive members of the series are considered. 
Sodium, for instance, acts on the alcohols with formation of sodium 

1 alkyl oxides and hydrogen according to the equation 

2ROH + 2Na = 2RONa + H 2 , 

but the vigour of the action is very different according as the alcohol 
is one high or low in the series. With methyl alcohol (CH 3 . OH) and 
with ethyl alcohol (C 2 H 5 . OH) the action is brisk ; with amyl alcohol 
(C 5 H n . OH) it is already sluggish at the ordinary temperature. 

Corresponding to this gradation of chemical activity within the 
c series we have a gradation in physical properties, and here, on account 
of the accuracy with which these physical properties can be measured, 
the differences are more readily observed and more readily brought 
under general rules. We may take first for consideration the specific 
gravities in the series of normal primary saturated alcohols, which 
i are exhibited in the following table (p. 128). The values of the specific 
gravity are for 0°, and are referred to the specific gravity of water at 0°. 

It will be seen that as the molecular weight of the alcohol increases, 
the specific gravity increases likewise. The difference in composition 
from step to step is one atom of carbon and two atoms of hydrogen ; 
and to this constant difference, CH 2 , there corresponds a continually 
diminishing difference in the values of the specific gravities as the 
series is ascended. 


Specific Gravity. 









C 2 H 5 .OH 


+ •011 



C 3 H 7 .OH 


+ •006 


» 5 

C 4 H 9 .OH 


+ •006 



C 5 H n .OH 


+ •004 


> J 

C 6 H 13 . OH 


+ •003 



C 7 H 15 .OH 


+ •003 


J ) 

CgHjf . OH 


+ •003 



C 9 H 19 . OH 


An exception is found in the first member of the series. Eeasoning 
by analogy from the other members, we should expect methyl alcohol 
to have a considerably lower specific gravity than ethyl alcohol, but 
instead of this it has a higher specific gravity, the value being inter- 
mediate between those for the second and third members of the series. 
The exceptional behaviour of the first member of a series is not 
confined to this series, or to this property, being of frequent occurrence 
amongst organic compounds. 

If instead of considering the specific gravities of the compounds 
{i.e. the weights which occupy unit volume) we consider the specific 
volumes (i.e. the volumes which are occupied by unit weight), or still 
better, the molecular volumes (i.e. the volumes occupied by the 
molecular weights), we are enabled to bring greater regularities to 
light. The specific volume v is the reciprocal of the density d, and 
the molecular volume V is the product of the specific volume and the 
molecular weight, or the molecular weight divided by the density. 

1 M 

The values of v = -~ and V= ~j are contained in the following table : — 






Methyl alcob 




+ 10 

+ 17.7 


C 2 H 5 OH 





+ 16-3 


C 3 H 7 OH 



- 9 


+ 16'5 


C 4 H 9 OH 



- 9 


+ 16-2 





- 5 


+ 16-4 


C 6 H 13 OH 



- 5 


+ 16-2 


C 7 H 15 OH 



- 4 


+ 16-2 


C 8 H 17 OH 



- 4 


+ 16-2 


, , C 9 Hj 9 OH 





The regularity exhibited by the molecular volumes is much more 

striking than that displayed by the specific volume or by the specific 

gravity. Here the difference between neighbouring members, instead 

of continuously diminishing, remains practically constant throughout 

the series. The volume occupied by the molecular weight of the 

alcohol is increased by 16*2 units for every addition of CH 2 to the 

molecule of the alcohol. The value 16*2 may therefore be looked 

upon as the " molecular " volume of CH 2 under the given conditions, 

and in this particular homologous series. Under other conditions the 

value for CH 2 may be, and is, different. The volume of organic 

I compounds is usually affected greatly by temperature ; the coefficient 

I of expansion of ethyl alcohol being, for example, some twenty times 

as great as that of water at the ordinary temperature. It is therefore 

of importance to determine under what conditions the volumes of 

different compounds are to be compared, more especially when they 

belong to different series. In the above instances the specific gravities 

were measured at 0° (and compared with water at 0°). This choice 

of temperature is evidently arbitrary, bearing no relation to the 

properties of the compounds themselves, but as far as we have seen 

it has the merit of leading to regular results. Kopp found, by 

studying a great many liquid substances, that if the molecular volume 

of each was determined at its own boiling point, not only were the 

g regularities within each series preserved, but the same regularity held 

t good for practically all series. No matter what homologous series 

j was studied, Kopp found that a difference of composition of CH 2 

: corresponded to a constant difference in the molecular volume, the 

I value being in his units 22. It must be noted that this volume is not 

absolutely constant, but is merely an average, the actual differences 

: being liable to slight fluctuations about the mean. The value is 

greater than that obtained when the homologous compounds are all 

•measured at the same temperature, because, as we shall see, the boiling 

points in homologous series rise as the series is ascended. Thus the 

molecular volumes of two neighbouring compounds measured at their 

1 respective boiling points will show a greater difference than if they 

were measured at the same temperature, for the molecular volume of 

the compound with greater molecular weight is ascertained at a higher 

temperature than the molecular volume of the substance with lower 

molecular weight, and there is therefore the expansion between the 

two temperatures to be added to the value that would be obtained if 

both were measured at the boiling point of the lower compound. 

Besides this regularity others come to light. It was found by 

Kopp that the densities of isomeric compounds (measured at their 

boiling points) were equal, and consequently that their molecular 

volumes were also equal under these conditions. For example, he 

found the following numbers for compounds having the formula 

C 6 H 12°2 : ~ 


Molecular Volume. 

Methyl valerate 149*2 

Ethyl butyrate 149*3 

Butyl acetate 149*3 

Amyl formate 149*8 

This constancy is not displayed if the measurements are made at the 
same temperature if the boiling points of the isomeric substances are 
widely different. Thus for the butyl alcohols we have, when the 
densities are all measured at 20° (against water at 4°) — 

Boiling Point. Density. 

Normal primary C 2 H 5 . CH 2 . CH 2 OH 117° 0*810 

Iso-primary (CH 3 ) 2 . CH . CH 2 OH 107° *806 

Tertiary * (CH 3 ) 3 .C.OH 83° 0*786 

The alcohol with highest boiling point has the greatest density, i.e. 
the smallest volume when the measurements are all made at one 
temperature. Its higher boiling point, however, allows of greater 
expansion, so that when the determinations are made at the boiling 
points, the increase of volume due to the higher temperature to some 
extent compensates for the smaller original volume at 20°. The 
choice of the boiling points of the compounds as the temperatures at 
which the comparisons are to be made, although it involves the 
properties of the compounds themselves, is still to a certain extent 
arbitrary, inasmuch as they are the temperatures at which all the 
substances have an arbitrary vapour pressure, viz. 76 cm. The 
justification of this choice lies in the fact that the observed regularities 
under these conditions are great; and although a better selection of 
conditions might conceivably be made, it is unlikely that any fresh 
regularities would be brought to light. 

The heat evolved by the complete combustion of an organic com- 
pound (the carbon becoming carbon dioxide, and the hydrogen becoming 
water) is an example of a property exhibiting constant differences 
between neighbouring members of a homologous series when molecular 
quantities are compared. The following table contains the heats of 
combustion of gram-molecular weights of the fatty acids expressed in 
centuple calories (p. 122) : — 

Acid. Difference. 

Formic CH 2 2 590 K 

Acetic C 2 H 4 2 2133 „ 

Propionic C 3 H 6 2 3679 ,, 

Butyric C 4 H 8 2 5227 „ 

Valeric C 5 H 10 O 2 6767 „ 

Caproic C 6 H 12 2 8312 „ 

For each difference in composition of CH 2 there is a difference in the 


molecular heat of combustion amounting on the average to 1543 K. 
This difference is found to be practically the same in all homologous 
series. Thus for the alcohols we have — 




CH 4 

1685 K 

1561 K 


C 2 H 6 

3246 „ 

1565 „ 


C 3 H 8 

4811 „ 

1565 „ 


C 4 H 10 O 

6376 „ 

1558 „ 


C 5 H 12 

7934 „ 

A property which in general varies regularly in homologous series 
is the boiling point. As we ascend a simple series the boiling point 
invariably rises, but the rise at each succeeding step in general grows 
smaller and smaller as the molecular weight increases. The following 
table gives the boiling points of some of the normal saturated 
hydrocarbons. In the second column under t is the boiling point at 
76 cm. in the centigrade scale; under T we have the boiling point in 
the absolute scale, i.e. t + 273. 



t (calculated). 



C 7 H 16 















C 10 H22 





QliH 24 















C 14 H 30 














The boiling points of most series exhibit a regularity similar to the 
above, but the differences are not usually so great as is the case with 
the hydrocarbons. The boiling points of most members of a homo- 
logous series can be expressed by a fairly simple formula. If M is the 
molecular weight of the compound, T its boiling point in the absolute 
scale, and a and b constants for the series, then in general 


The values under " t calculated " in the above table were obtained by 
means of a formula of this kind, the constants for the series being a = 


37*38 and 5 = 0*5. In some series (e.g. the alcohols, the alkyl 
bromides, and the alkyl iodides) a formula of this type cannot success- 
fully be applied, but in most cases it gives accurate results. 

The student must again be reminded that the boiling point of a 
series of compounds is a magnitude arbitrarily selected in so far as the 
pressure under which the compound boils is itself quite arbitrarily 
chosen equal to the average pressure of the atmosphere. It is found, 
however, that a formula of the above type is capable of expressing the 
relation between boiling points and molecular weights under any 
pressure. In the same series the constant a has different values for 
different pressures, whilst the constant b retains the same value for 
all the pressures. Thus the boiling points of the above series of 
hydrocarbons under 3 cm. pressure may be expressed by the formula 

r=29*68ilf ' 5 , 

instead of 

r=37*38if ' 5 , 

which is valid for 76 cm. The constancy of b for different pressures 
leads to the following results. If we take two substances belonging to 
the same series, we have for their boiling points at a certain pressure p 

T=a3P, and T x = aM*. 

At another pressure p' we have the boiling points 

T^dM*, and T{ = a!M*. 

M, M and b remain the same throughout; so, by dividing each 
equation of the first pair by the corresponding equation of the second 
pair, we obtain 

-^ and -?,= ? whence F = ^. 

That is, if b remains constant for different pressures, the ratio of the 
boiling points (expressed in the absolute scale) at any two given 
pressures will remain the same for all members of the homologous 
series. Transposing the last equation, we have 

i.e. the ratio of the absolute boiling points of two substances belonging 
to the same homologous series is independent of the pressure. 

The boiling points of isomeric substances are in general not the 
same, as may be seen, for example, in the case of the butyl alcohols 
given in the table on p. 130. We usually find, as here, that the 
isomers containing the longest carbon chain boil at a higher temper- 
ature than those with branched carbon chains. 

As a rule, the boiling point of the first member of a homologous 




series is considerably higher than that calculated from the formula 
which includes the other members of the series. This abnormally 
high boiling point is displayed still more markedly when, instead 
of one characteristic group, the first member of the series has two. 
Thus, for example, in the simplest series of the dicyano-derivatives, 
the first member has a boiling point which is actually higher than 
those of the three succeeding members : — 

+ 9 
+ 10 

Malonic nitrile (CN) 2 CH 2 218° 

Methyl-malonic nitrile (CN) 2 CH . CH 3 197° 

Ethyl-malonic nitrile (CN) 2 CH . CH 2 CH 3 206° 

Propyl-malonic nitrile (CN) 2 CH . CH 2 CH 2 CH 3 216° 

The glycols behave similarly : — 

Ethylene glycol CH 2 (OH) . CH 2 (OH) 197° 

Methyl-ethylene glycol CH 2 (OH) . CH(OH) . CH 3 188° 

Ethyl-ethylene glycol CH 2 (OH) . CH(OH) . CH 2 CH 3 192° 

The melting points in homologous series often show the peculi- 
arity that the substances with an even number of carbon atoms form 
a regular series by themselves, and those with an odd number of 
carbon atoms form a regular series by themselves. The following 
table contains the melting points of the higher fatty acids : — 

+ 4 

Acid (E 


Melting Point. 


I (Odd). 


C 6 H ]2 2 



C 7 H 14 2 




+ 16-5 

+ 12*5 




^10** 20^2 

+ 31-4 

+ 28 





+ 44 

+ 40*5 





+ 54 

+ 51 





+ 62 

+ 60 




^18 "36^2 

+ 68 

+ 66'5 

Ci9H 3 g0 2 




+ 75 

If we take the successive members of the series, we find an alternate 
rise and fall in the melting point as we pass from one acid to the 
next. If, however, we separate the acids into those with an even 
and those with an odd number of carbon atoms, we have a rise in 
the melting point in each series. It will be observed here again 
that the differences decrease as we ascend the series. 

In some homologous series the difference between the homologues 
with even and those with uneven numbers of carbon atoms is so 



marked that in the one case we may have the melting point rise, 
and in the other case fall, as we ascend. The normal saturated 
dibasic acids afford an instance in* point : — 

Acid (Even). 

Melting Point. 

Acid (Odd). 


C 4 H 6 4 



C 5 H 8 4 Glutaric 


C H 10 O 4 



C 7 H 12 4 Pimelic 


C 8 H 14 4 



C 9 H 16 4 Azelaic 1 


CioH 18 4 






C13H24O4 Brassylic 

Dodecane-dicarboxylic C 14 H 26 4 


In the even series the melting point falls, in the odd series the melting 
point rises, as the molecular weight increases. Once more the rise or 
fall diminishes in magnitude step by step as we ascend the series. 

The student may have observed that in these melting point tables 
the lowest members of the series have been omitted. This is so because 
they do not fall under the general scheme which includes the higher 
members of the series. For example, the first member of the normal 
dibasic acids, oxalic acid, C 2 H 2 4 , melts at 189°, and the second 
member, malonic acid, C 3 H 4 4 , melts at 133°. It is evident that the 
melting points are not included in the general scheme which suffices 
for the other members of the series. In the series of the fatty acids the 
same irregularity is displayed. Acetic acid, C 2 H 4 2 , which, if the fall 
observable as we descend the series were maintained to the end, should 
melt many degrees below zero, in reality melts at + 16*5°. In general, 
we may say that the lowest member (or members) of a series departs from 
the regular behaviour exhibited by the higher members amongst them- 
selves. Exceptions to this rule occur, but they are comparatively rare. 

The separation of the members of a series into those with even 
and those with odd numbers of carbon atoms sometimes appears in 
other properties besides the melting points. Thus in the same series 
of the normal dibasic acids we have the solubility of the odd 
members in water considerably greater than the solubility of the 
even members, as the following table shows : — 

Acid (Even). 


Acid (Odd). 

Oxalic C 2 H 2 4 



C 3 H 4 4 Malonic 

Succinic C 4 H 6 4 


C 5 H 8 4 Glutaric 

Adipic C 6 H 10 O 4 



C 7 H 12 4 Pimelic 

Suberic C 8 H 14 4 



C9H 16 4 Azelaic 1 

Sebacic C 10 H 18 O 4 


1 It is still somewhat doubtful if azelaic acid has the normal structure. 


The solubilities are given as parts of acid dissolved by 100 parts 
of water at the ordinary temperature (15° - 20°). It will be seen that 
in the separate series the solubility in water falls off very rapidly as 
the number of carbon atoms in the molecule increases. This is in 
accordance with what was stated in the chapter on solubility. The 
solubility of an acid in water is probably connected with the presence 
in it of the hydroxyl ( - OH) or carboxyl ( - COOH) group (cp. 
p. 56). Other things being equal, the greater the proportion of 
hydroxyl (or carboxyl) in the molecule, the greater will be its solu- 
bility. As we go up the series the proportion which the hydroxyl 
bears towards the rest of the molecule diminishes, and along with this 
goes on diminution of the solubility in water. 

It is evident from the above tables that in this particular series 
there is some fundamental difference between the homologues with an 
even and those with an odd number of carbon atoms. This difference is 
probably to be sought for in some property affecting the substance in 
the solid state, for both the melting point and the solubility are here 
properties of the solids, i.e. it is the presence of the solid that deter- 
mines both. The liquid may be supercooled when not in contact with 
the solid, and the solution may be supersaturated if the solid is not 
present. But the solid cannot be heated above its point of fusion 
without melting, and the solution in contact with it is always exactly 
saturated. It is in this sense that we say that it is the solid that 
determines the melting point and the solubility — not the liquid or 
the solution. 

The analogy that we see between solubility and fusing point in 
the above tables is an example of a rule of fairly general applicability. 
We usually find that, when we compare similar substances, fusibility 
and solubility go together. If we consider a set of isomeric substances, 
for instance, we find that the order of solubility is usually the same as 
the order of fusibility, i.e. the most soluble isomer has the lowest 
melting point. 

The order of the solubility of isomers is frequently independent 
of the nature of the solvent. Thus, if one of two isomers is more 
soluble in water than the other, it will still be the more soluble if 
alcohol, ether, benzene, etc., be the solvents employed instead of water. 
In some special cases, not only the order but even the ratio of the two 
isomers remains nearly the same for all solvents. For example, it has 
been found that meta-nitraniline is, on the average, 1*3 times more 
soluble than para-nitraniline in 13 different solvents, the ratio of 
solubility only varying from 1*15 to 1*48. It has also been found 
that the order of solubility of corresponding salts of isomeric acids is 
also very frequently the order of solubility of the acids themselves. 
It must be borne in mind, however, that these rules are all liable to 
well-marked exceptions. 



The properties of substances, when studied in relation to their 
composition and structure, have been divided into three classes. In 
the first class we have those properties which are possessed by the 
atoms unchanged, no matter in what physical or chemical state these 
atoms may exist. Such properties are called additive, and the best 
instance of an additive property is found in weight (or mass). Each 
atom retains its weight unaltered whether it exists in the free state 
or whether it is combined with other atoms. When atoms com- 
bine, the weight of the compound is the sum of the weights of the 
component atoms. This is only another way of stating one of 
the fundamental assumptions of the atomic theory (Chap. II.), the 
assumption, namely, which takes account of the indestructibility of 
matter. Weight is the additive property par excellence, no other 
property being additive in the strict sense, although in the case of 
some properties there is an approximation to the additive character. 

We have seen in homologous series that there exists between 
neighbouring members a difference in molecular volume which is 
practically constant. For a difference in composition of CH 2 there is 
the constant difference of 22 in the value of the molecular volume, 
which retains nearly the same value for all homologous series. We 
may therefore attribute the value 22 to the group CH 2 , for whenever 
the methylene group enters a molecule the molecular volume is 
increased by this amount. Here we are evidently dealing with an 
additive property, but the additive character is modified by other 
influences, for the difference 22 is not absolutely constant, but 
fluctuates slightly about this value. It should be noted, also, that 
this number only holds good for the liquid state, for the measurements 
from which it is derived were all made at the boiling points of the 
liquid substances. By comparing the molecular volumes of liquids 
differing in composition in various definite ways, Kopp was able to 
establish a set of approximate rules such as the following : — 


(a) When two atoms of hydrogen are replaced by one atom of 
oxygen, there is a very slight increase in the molecular volume. 

(b) One atom of carbon may replace two atoms of hydrogen without 
sensible alteration of the molecular volume. 

From these rules, in conjunction with the preceding one, we may 
draw the following deductions : If the increase of molecular volume 
for CH 2 is 22, and if one atom of carbon is equivalent to two atoms 
of hydrogen, we may assign the value 11 to an atom of carbon, and 
the value 11 -f- 2 = 5*5 to the atom of hydrogen. These values, then, 
are assumed to be the atomic volumes of carbon and hydrogen. 
Since there is a slight increase in the molecular volume when two 
atoms of hydrogen are replaced by one atom of oxygen which is 
attached to the same carbon atom, the atomic volume of oxygen must 
be somewhat greater than 11. On the average it is 12*1. It is 
found, however, that when hydrogen is replaced by hydroxyl, the 
increase of molecular volume is not 12*1, corresponding to the 
addition of one oxygen atom, but only about 7*8. When oxygen, 
therefore, is attached to one carbon atom, it contributes more to the 
molecular volume than when it is partially attached to carbon and 
partially to hydrogen. Here we come across an influence which 
modifies the additive character of all properties except weight, namely, 
the influence of structure or constitution. The molecular volume is 
not a purely additive property — it is in part constitutive, i.e. is 
dependent not merely on the number and kind of atoms in the mole- 
cule but also on their arrangement. We must therefore attribute to 
oxygen two atomic volumes — 12*2 when it is attached to carbon so as 
to form the carbonyl group (CO), and 7*8 when it is part of the 
hydroxyl group, or is attached to two different carbon atoms, as in the 

We are now in a position to deduce the molecular volume of a 
compound containing only carbon, hydrogen, and oxygen, by adding 
together the volume values of its constituent atoms. Thus, a com- 
pound of the formula C a H&O c O'tf, where 0' denotes oxygen in a 
hydroxyl group, has the molecular volume 

F=lla + 5*5&+12*2c + 7*8d, 

if the molecular volume is determined at the boiling point of the 
liquid. For example, in valeric acid, C 4 H 9 . CO . OH, we have a = 5, 
b = 10, c = 1, d = 1, so that 

V= 55 + 55 + 12-2 + 7-8 = 130. 

The molecular volume as experimentally ascertained is 130*5. The 
agreement in the majority of cases is not so good, e.g. the molecular 
volume of ethyl oxalate found from the formula is 161, and that 
found by experiment 167. In this connection it must be remembered 


that Kopp's original rules are not strictly accurate, so that the 
deductions from them are always liable to slight error. 

From the consideration of compounds containing other elements 
than those already referred to, values have been deduced for the 
atomic values of nitrogen, the halogens, sulphur, phosphorus, etc. 
Sulphur and nitrogen, like oxygen, have different values for the 
atomic volume according to the mode in which they are combined 
with other atoms. 

When an element or radical exists in the liquid state, the mole- 
cular volume deduced from its compounds in general agrees with that 
of the free element or radical. Thus the volume of Br 2 deduced from 
bromine compounds would be 53*4, and the molecular volume of free 
bromine is actually 53*6. The value of N0 2 deduced from compounds 
containing it as a radical is 31*5, the value for the free oxide being 

No purely additive property can throw any light on the size of 
the molecule or its constitution, as each atom retains the numerical 
value of the property unchanged whether it exists in the free state 
or whether it is combined with other atoms in any way whatever. 
As the value is therefore unaffected by the kind and extent of the 
combination, it can clearly give no indication as to what that mode or 
extent of combination may be. When the property is modified by 
constitutive influences, as is the case with the molecular volume of 
liquids, it may throw light on the constitution of a substance. Thus, 
if of two isomeric substances it were suspected that one contained an 
atom of carbonyl oxygen, whilst the other contained an atom of 
hydroxyl oxygen, that with the larger molecular volume would be the 
compound containing the hydroxyl oxygen. 

The refractive power of liquids is a property which, like the 
molecular volume, is in general additive in character, although modi- 
fied by constitutional influences. The refractive index itself cannot 
be taken as a measure of the refractive power when its relation to the 
chemical nature of the substance is under investigation, for the index 
varies greatly with temperature, etc. A better measure is found in 

the specific refractive constant , , or (n - l)v, where n is the refrac- 
tive index, d the density, and v the specific volume. This expression 
varies very slightly with the temperature, and is little influenced by 
the presence of other substances, so that it has frequently been used 
in the comparison of different liquids. Another specific refractive 

constant is given by the expression — z — - . i,or -^ — - . v, which was 

arrived at on theoretical grounds. When the values obtained by its aid 
are compared with the values of (n - l)v, it is found that they have an 
advantage over the latter, inasmuch as they are not only independent 
of the temperature, but also of the state of aggregation. With the 


empirical refraction constant there is considerable divergence between 
the values in the liquid and gaseous states, whilst the numbers 
obtained for the theoretical constant are the same in both cases. Thus 
for water at 10° we have 

, ., . rfi - 1 

(u - 1)y ^T2* V 

Liquid .... 0*333S 0*2061 

Gaseous .... 0*3101 0*2068 

The molecular refractive power is the product of the mole- 
cular weight into the specific refractive power as measured by either 
of these expressions. Thus we have the " empirical " molecular refrac- 
tion, Mv(n - 1), or F(n - 1), and the " theoretical " molecular refraction, 

n 2 - 1 n 2 - 1 

— — - . Mv, or —Z — - . V. It is found that the molecular refraction of 

n A + 2 n 2 + 2 

liquids as measured by either of the formulae is essentially an additive 
property modified by constitutive influences, so that an atomic refrac- 
tion for carbon, hydrogen, oxygen, chlorine, etc., may be calculated. 
If the refractive powers of each atom in the molecule be added 
together, the sum gives the molecular refraction of the compound. 
As is the case with the atomic volumes, different values have to be 
attributed to the atomic refraction of oxygen, according as it is 
carbonyl, or hydroxyl, or ether oxygen, a distinction in this case 
being necessary between the last two kinds which is not required 
for atomic volumes. When a substance contains an ethylene linkage, 
its atomic refraction is considerably higher than would be reckoned 
from the atomic refractions of the elements composing it ; and when 
an acetylene linkage is known to exist in the molecule, the excess is 
even higher. "Atomic" refractions have therefore been attributed to 
"double bonds" and "triple bonds," which must be added to the 
atomic refractions of the elements themselves when the total molecular 
refraction is calculated. 

Some liquids exhibit the phenomenon of optical activity, that is, 

when placed in the path of a polarised ray they rotate the plane of 

polarisation in one sense or the other. Substances which rotate the 

plane of polarisation in the direction of the hands of a watch are said 

to be dextrorotatory; substances which rotate it in the opposite 

direction are said to be bevorotatory. The specific rotatory power 

is usually denoted by the symbol [a], which is obtained by dividing 

the actual rotation observed in the polarimeter by the length of the 

• layer of liquid through which the light passes, and by the density 

. of the liquid at the temperature of observation. To obtain convenient 

. numbers, the length is usually given in decimetres. The molecular 

rotation is the product of the specific rotation into the molecular 

. weight, or more usually the hundredth part of this value, i.e. 

r 1 Ma a 



where a is the observed rotation, I the length of liquid, and d its density. 
It has been suggested that a better expression would be [8] = - V V, since 


we are obviously here concerned not so much with the molecular 
volume of the liquid as with the average distance between the centres 
of the molecules, the ray of light passing in a straight line through 
the liquid, and meeting a number of molecules proportional to the 
cube root of the molecular volume. Since the rotatory power varies 
with the temperature and with the wave length of the light employed, 
it is necessary to specify both of these in stating the value of the 
specific or molecular rotation. 

When we come to inquire into the nature of liquids and sub- 
stances in the dissolved state which show optical activity, we find that 
they possess in every case one or more asymmetric carbon atoms, i.e. 
carbon atoms which are united to four kinds of elements or radicals 
all different from each other. If we imagine these four different 
radicals to be at the four corners of a tetrahedron, we find that we 

can arrange them in two essentially different ways, as is shown in 
the accompanying figures. Here the tetrahedra are supposed to be 
resting with one face on the paper, the summits being towards the 
reader. If we call the four different groups a, b, c, d, and place the 
group d at the summit, then in Fig. 22 the order abc is in the 
direction of the hands of a watch, and in Fig. 23 in the reverse 
direction. If two of the groups are made the same, the asymmetry 
vanishes, as we can see from the figures if we make b = c, the two 
figures then becoming identical. No altogether satisfactory answer 
has yet been obtained to the question of what determines the value 
of the molecular rotation in any particular case. As we have seen, 
the rotation should vanish if two of the groups become identical, and 
it may also be perceived from the figures that if we interchange the 
positions of any two of the groups, the sign of the rotation will 
thereby be changed. If we suppose each of the groups to be en- 
dowed with some definite property causing the rotation, and denote 
the value of this function by a, /?, y, 8 for the groups a, b, c, d, 
respectively, the rotation of the asymmetric carbon atom must be 


determined by an expression of the following or similar type: — 
(a -/})(/? - r )( y -S)(a- y )(a-S)G8-S). 

If the function becomes the same for two of the groups, the expression 
becomes zero, i.e. the rotation vanishes, and if we interchange any two 
throughout, the sign of the whole expression is changed, i.e. the rota- 
tion from dextrorotatory becomes lsevorotatory, or vice versa. What 
the function is we do not know. It seems to be connected with the 
weight of the radicals, but cannot be the weight itself, since substances 
are known to be optically active which have two groups of equal 
weight attached to the asymmetric carbon atom. Certain regularities 
have been observed among the molecular rotations of members of 
homologous series, but numerous exceptions occur. Many of the ap- 
parent divergencies, however, may be accounted for by the assumption 
that different members of the same series may have different degrees 
of molecular complexity when in the liquid state (cp. Chap. XIX.). 

Some crystalline substances, such as quartz, are optically active, 
but the activity here is not due to the arrangement of atoms within 
the molecule, but rather to a certain arrangement of the crystalline 
particles. A consequence of this is that the activity disappears when 
the crystalline structure is destroyed, i.e. when the substance passes 
into the fused or dissolved state. As a general rule, substances which 
are active in the liquid or dissolved state are not active when 
crystalline, but a few substances are known which exhibit optical 
activity both as crystals and as liquids. 

All liquids when placed between the poles of a magnet or in the 
core of an electromagnet become optically active. The character of 
the magnetic optical activity, however, is essentially different from 
that exhibited by liquids which are naturally active. If we place a tube 
containing a naturally active liquid between the prisms of a polarising 
apparatus, and so adjust the prisms that no light passes through the 
system when we place a light-source at one end and the eye at the 
other ; and if we then reverse the positions of the light-source and 
the eye, we find that the passage of the light is still obstructed. The 
sense and magnitude of the activity, then, are independent of the 
direction of the light. A naturally inactive liquid in a magnetic field 
behaves quite differently. If we first adjust the prisms so that no 
light passes, and then reverse the positions of eye and light-source, 
we find that light now traverses the system quite freely. We find, 
in fact, on readjusting the prisms to darkness, that a substance which 
appeared originally dextrorotatory is now to an equal extent 
laevorotatory. The difference in the nature of the activity may best 
be made clear by analogy. The action of a naturally active liquid 
resembles the action of a screw. If a screw is right handed 
(dextrorotatory) when viewed from one end of its axis, it is right 
handed when viewed from the other. A naturally active liquid if 


dextrorotatory when viewed from one end remains dextrorotatory 
though the direction of the light passing through it is reversed. The 
polarity of a magnetically active liquid resembles that of a muff. If 
the lie of the hair in a muff when viewed from one end of its axis is 
in the direction of the hands of a watch, it will be in the opposite 
direction if we view it from the other end of the axis. A magnetically 
active liquid is similarly dextro- or lsevo-rotatory according to the 
direction in which the light passes through it, the magnetic field 
being supposed constant. 

The specific magnetic rotation is usually given as the ratio of 
the specific rotation of the substance (determined as in naturally active 
liquids) to that of water under the same conditions: the molecular 
magnetic rotation is this value multiplied by the molecular weight of 
the substance and divided by 18, the supposed molecular weight 
of water. We find once more that when the molecular magnitudes 
for homologous substances are compared, there is a constant difference 
for the group CH 2 , so that to this extent the property is an additive 
one. The constitutive influence, however, is much more marked here 
than in any of the previous instances ; and the property is therefore 
valuable when applied to solving problems of constitution. 

When a relation has been established between the constitution of 
well-known substances and the values of a certain property possessed 
by them, it is legitimate to draw inferences regarding the un- 
known constitution of other substances from the known value of 
this property in their case, the same rules being applied as to 
substances of known constitution. Of course, the worth of the 
deduction depends entirely on the number and variety of compounds 
which have been investigated, and on the exactness of the empirical 
rules established from the investigation. Melting points and boiling 
points, for example, are occasionally useful in indicating the probable 
structure of a compound, but it is seldom that the evidence based on 
them alone is to be treated with any degree of confidence. To take 
an instance in point, it was assumed that the decane dicarboxylic 
acid given in the table of melting points on p. 134 has the normal 
structure, entirely on the strength of its' melting point falling into 
the regular scheme displayed by the even members of the homo- 
logous series which are known to have the normal structure ; for if 
its structure were not normal, it would in all probability have a 
melting point diverging widely from the scheme which included the 
members of the series which were known to be normal in reality. 
This evidence is slight, but for want of anything better it has a , 
certain validity, although any well-established fact to the contrary 
would at once be conclusive against it. 

When the connection between the constitution and the value of a 
physical property is better marked and susceptible of being laid down 
in more definite rules, as is the case with the molecular refraction or 


magnetic rotation, greater confidence can be placed in the conclusions 
drawn from the value of the property in a particular compound as to 
the constitution of that compound. Great care, however, must be 
exercised in certain cases, for we are liable to draw conclusions which 
are not warranted by the facts. Thus, for example, from a considera- 
tion of the molecular refraction of aromatic compounds, it has been 
concluded that benzene has three ethylene linkings in the molecule, 



HC f / ^7i CH 

i.e. that Kekule's formula H J IL H is the correct one. Against this 


we must place the fact that benzene does not behave chemically as if 
it had a molecule containing three ethylene linkings, and at present 
the chemical evidence must be held to outweigh the evidence of the 
molecular refraction. The real difficulty in a case like this is that 
we can express the general chemical behaviour of the fatty compounds 
by means of three different kinds of carbon linking — simple, ethylenic, 
and acetylenic, all perfectly well denned chemically ; whilst in the 
aromatic compounds we meet with something entirely new, and not 
readily brought into any of the above classes of carbon linking. How 
we are to represent this new kind of linking we do not at present 
know. Various attempts have been made, mostly based on the 
supposed properties of the tetrahedral carbon atom — some of them 
yielding ordinary static formulae, some of them kinetic formulae in 
which the carbon atoms are assumed to be in a state of constant 
vibration in a specified manner. These formulae have all their 
peculiar merits and demerits. None can be held to be entirely 
satisfactory, in view of the conflict of evidence, and it may perhaps 


still be said that the least definite formula i , which makes no 

hc^ . yen 

assumptions as to the mode of linking of some of the bonds, is also the 
happiest in representing the known properties of aromatic compounds. 
As we have seen in Chapter XIII., the molecular heat of combustion 
is essentially an additive property, though subject to modifications 
conditioned by differences in constitution. The modifications occa- 
sioned by changes of constitution are often very slight, so that as long 
as we are dealing with saturated compounds, it is found that isomeric 
substances have approximately the same heats of combustion. Thus 
for propyl alcohol we have 4986 K, and for isopropyl alcohol 4933 K ; 


for anthracene and phenanthrene, which both have the formula C U H 10 , 
we have 16,943 K and 16,935 K respectively. In other cases the 
differences are greater, but when substances of nearly the same 
character are considered, the differences are not of much importance. 
Unsaturated compounds exhibit great divergence from saturated 
compounds, and values have been attributed to the ethylene linkage 
and to the acetylene linkage. Conclusions, too, have been drawn as 
to the constitution of benzene from the heat of combustion. This 
amounts for benzene, C 6 H 6 , to 7878 K, the value for the isomeric 
substances, dipropargyl, CH • C . CH 2 . CH 2 . C • CH, and dimethyl- 
diacetylene, CH 3 . C • C. C \ C . CH 3 , being 8829 K and 8474 K 
respectively. Here the differences for the isomeric substances are 
great, owing to undoubted differences in the constitution — especially 
in the mode of linking of the carbon atoms. The exact nature of 
the differences in constitution corresponding to the differences in the 
heat of combustion is, however, as before, uncertain, for we have 
nothing but analogy to guide us, and are apt to assume that the 
rules that hold good for the fatty compounds have the same validity 
for aromatic compounds, which in all probability is not the case. 

Some properties are exceedingly valuable in certain special cases 
for giving us an insight into the constitution of chemical compounds. 
For instance, if a new compound displays optical activity, we know 
that in all likelihood it contains one or more asymmetrical carbon 
atoms, i.e. carbon atoms which are combined with four different kinds 
of atoms or groups of atoms ; for all well-investigated optically active 
compounds have been proved to contain such asymmetric carbon 
atoms. Again, for organic acids we have the molecular conductivity 
in aqueous solution (cp. Chap. XXI). This property, as we shall see, 
is closely related to the strengths of the acids, and also to their 
constitution. The molecular conductivity of acids varies with the 
strength of the solution in which they exist, and the actual 
variation is different according to the acid which is dissolved. 
There is a general rule, however, to which the variations for nearly 
all acids conform, and this enables us to calculate for each acid a 
constant independent of the strength of the solution whose conductivity 
is measured. It is this dissociation constant which is useful in the 
discussion of questions regarding the constitution of organic acids. In 
it there is scarcely a trace of an additive nature in the sense in which 
the term has hitherto been used. For example, the dissociation 
constants K for the fatty acids are — 







CH 3 




C 2 H 5 




C 3 H 7 




C 3 H 7 . 








Q5H11 . 




If we except the first number of the series, which as usual diverges 
from the others, we see that the constants have all approximately the 
•same values, although constant additions are made to the molecule. 

In the case of the normal dibasic acids of the oxalic series we have 
a very different table : — 


Oxalic (COOH) 2 10 -0 (?) 

Malonic CH 2 (COOH) 2 0*163 

Succinic C 2 H 4 (COOH) 2 0-00665 

Glutaric C 3 H 6 (COOH)o 0'00475 

Adipic C 4 H 8 (COOH) 2 0*0037 

Here the values of the constants sink steadily as the molecular 
weight increases, although the fall becomes less and less as we proceed. 
In the normal acids the carbon atoms are supposed to be linked to 
each other in a continuous chain. The two carboxyl groups are there- 
fore at opposite ends of the chain. Now, in the fatty monobasic acids, 
in which there is only one carboxyl group, the addition of CH 2 has 
little influence on the dissociation constant. We therefore attribute 
the great diminution of the value in the series of dibasic acids to the 
increasing distance of the carboxyl groups from each other. In oxalic 
acid the carboxyl groups are directly attached, and their proximity is 
assumed to increase the dissociation constant. Each separate group 
has acid properties, and the two groups reinforce each other's acidity. 
In a higher member of the series, e.g. adipic acid, it is true that we 
have still two acid groups, but they are so far apart that we suppose 
them to have little reciprocal influence in increasing the acid properties. 
This mode of viewing the relation between constitution and dissocia- 
tion constant leads on the whole to consistent results where the con- 
stitution has been well ascertained. 

When chlorine replaces hydrogen in an organic acid, the strength 
of the acid, and with it the dissociation constant, is increased. Thus 
trichloracetic acid is very much stronger than acetic acid from which 
it is derived, being comparable in point of strength with the ordinary 
mineral acids. The dissociation constants of the chloracetic acids are 
shown in the following table : — 






CH 2 C1 . COOH 
CCl 3 .COOH 


The replacement of hydrogen by chlorine obviously does not correspond 
to a constant addition to the dissociation constant, but rather to 
multiplication by a factor, although here the factor at successive stages 

The influence of other replacing atoms and groups is seen in the 
following derivatives of acetic acid : — 






CH 2 (0H) . COOH 


CH 2 (SH) . COOH 










CH 2 (CN) . COOH 





From the table it appears that the influence of a chlorine atom is 
almost as great as the influence of another carboxyl group, although 
much less than the influence of a cyanogen group. Eeasoning on the 
same lines as those adopted in treating the constants of the normal 
dibasic acids, we should expect that the farther removed any of these 
substituting atoms or radicals are from the carboxyl group, the less 
will be their influence on the dissociation constant. This expectation 
is in general justified. We have, for example, the following constants 
for the hydroxyl derivatives of propionic acid : — 

Acid. Formula. K. 

Propionic CH 3 . CH 2 . COOH -00134 

jS-Lactic CH 2 (OH) . CH 2 . COOH 0-00311 

a-Lactic CH 2 . CH(OH) . COOH 0*0138 

Glyceric CH 2 (OH) . CH(OH) . COOH -0228 

The influence of the hydroxyl group in the a-position is much more 
marked than the influence of the same group in the /^-position. When 
there is a hydroxyl group attached to each carbon atom, the influence is 
greater still. 

In the aromatic series we meet with peculiarities not easily explic- 
able. Benzoic acid, C 6 H 5 . COOH, has the constant 0*0060, whilst the 
three hydroxyl acids have the values given below : — 


i J i 

H-C(J/C— OH H— C^'/C— H H— C^'yC-H 

H-c(, yC-H H-Cv |")c— OH H— C^fiyC— H 

C C C 

I I I 


Salicylic. Metahydroxy benzoic. Parahydroxybenzoic. 

K = 0*102 0-0087 0*0029 

In the orthohydroxyl acid, where the carboxyl and hydroxyl groups 
are on neighbouring carbon atoms, we have a great increase over the 
value of the constant of the parent acid. In the meta acid the effect 
on the constant is comparatively slight. In the para acid, contrary to 
expectation, we have an actual diminution instead of an increase. It 
is obvious from a consideration of these results that very great caution 


must be exercised in drawing conclusions for aromatic bodies from rules 
derived from the study of fatty compounds. 

The so-calledgeometrical or space isomerism is often accompanied with 
striking differences in the values of the dissociation constants. From 
the saturated dibasic acid, succinic acid, COOH . CH 2 . CH 2 . COOH, are 
derived two unsaturated acids, maleic acid and fumaric acid, which both 
in all probability have the structural formula COOH • CH : CH • COOH. 
We often express the difference between these two acids by writing 
their formulae thus : — 


II It 


Maleic acid. Fumaric acid. 

K = 1'17 K = 0'093 

the difference corresponding to a supposed difference in the arrange- 
ment of the hydrogen and carboxyl groups on the carbon tetrahedra. 
j The dissociation constants are much greater than that of succinic acid, 
and the constant of maleic acid is twelve times that of fumaric acid. The 
above mode of formulation might lead us to expect such a difference, 
for in maleic acid the two carboxyl groups are on the same side of the 
central plane, and in fumaric acid on different sides, and consequently 
more remote from each other. 

From the purely chemical point of view, the investigation of the rela- 
tionship between the value of physical properties and the constitution 
of chemical compounds is chiefly important as affording a means of 
ascertaining the otherwise unknown constitution of certain compounds 
by a determination of their physical constants. Too much reliance, 
however, should not be placed on this mode of settling chemical 
constitutions. The interpretation of the results is often doubtful, 
and that most frequently in cases where the determination of the 
constitution by chemical methods presents special difficulty. To 
sum up, we may say that the physical method usually offers valuable 
indications of the direction in which a chemical solution of the problem 
is to be sought, rather than a final solution of the problem itself. 

Besides additive and constitutive properties, we have what have 
been called colligative properties. The numerical value of these 
properties depends only on the number of molecules concerned, not 

Ion their nature or magnitude. For example, if we take equal numbers 
of gaseous molecules of any kinds whatever, they will always occupy 
the same volume when under the same conditions (Avogadro's Law). 
| The gaseous volume then is a colligative property. We make use of 
these properties in the determination of molecular weights, and they 
will therefore be further referred to under that heading. 



If we ask ourselves the question : " To which state of aggregation is the 
state of a substance in solution comparable 1 " we find that there are 
only two answers admissible, viz. the liquid state or the gaseous state. 
It is obvious that when a solid is dissolved in a liquid it at once loses 
the properties which are characteristic of the solid state. Its particles 
become mobile, and all properties which depend on regular arrange- 
ment of particles disappear. Thus the solid may be a double-refracting 
crystal; its solution exhibits none of the phenomena of double 
refraction. It may be an optically active solid, and yet its solution 
may show no signs of optical activity. In such cases the passage of 
the substance into solution exhibits considerable analogy to the passage 
of the substance into the liquid state. A double-refracting crystal 
almost invariably loses its double refraction when it melts, and most 
substances which are optically active in the crystalline state are inactive 
after fusion. The analogy which here holds good between the 
dissolved and the liquid states might, however, be equally well applied 
to the dissolved and the gaseous states. It is true that the solution 
of a substance in a liquid solvent is itself a liquid, but it by no means 
follows that the state of the substance within the solution is accurately 
comparable to that of a liquid. Indeed, if we look a little more 
closely into the matter we find that in the case of dilute solutions, at 
least, there is far more likelihood of the dissolved substance being in 
a condition comparable with that of a gas. 

One of the characteristic properties of a gas is its power of 
diffusion. If its pressure (or what is proportional to its pressure, its 
density or its concentration) is greater at one part of the space 
containing it than at another, the gas will move from the region of 
higher to the region of lower pressure or concentration, until by this 
process of diffusion the pressure or concentration is everywhere 
equalised. This process of diffusion goes on independently of the 
presence of another gas. A coloured gas such as bromine vapour may 


be seen to diffuse against gravity into another gas, say air, until the 
colour of the contents of the cylinder is everywhere of the same 
depth. The rate at which the diffusion takes place is, however, 
greatly influenced by the presence of another gas, the rate becoming 
rapidly less as the concentration of the other gas increases. This may 
be easily rendered evident by taking two cylinders, one evacuated, 
and one containing air, and breaking a bulb containing liquid bromine 
at the bottom of each. In the cylinder containing air the diffusion 
takes place slowly, an hour perhaps elapsing before the bromine vapour 
reaches the top of the vessel. In the evacuated cylinder the diffusion 
is apparently instantaneous. The particles of the foreign gas thus 
obstruct the movement of the particles of bromine, and render the 
process of diffusion slower. 

Now, in the case of a substance in solution we have the same 
process of diffusion as we have with gases. If we take a solution of 
a coloured substance, such as bromine itself, or preferably a coloured 
salt like copper sulphate, and place it in the bottom of a cylinder, 
afterwards covering it with a layer of pure water, we find that the 
colour of the copper sulphate solution gradually rises in the cylinder, 
proving that the copper sulphate is moving from a region of greater 
concentration to a region of less or no concentration. In this case 
also the diffusion is against gravity, for the solution of copper sulphate 
if concentrated has a much higher specific gravity than water, so that 
the centre of gravity of the liquid will be raised as the copper 
sulphate becomes more uniformly distributed throughout it. 

It is true that the diffusion of a dissolved substance is very much 
I slower than the diffusion of a gas, months or even years elapsing before 
I uniform concentration is attained in a cylinder not more than a foot 
high. But the difference is only a difference in degree, for it has 
been shown that gases under great pressures mix with extreme 
slowness against the action of gravity, many of the characteristic 
phenomena of the critical point, where the pressure is high, being 
obscured owing to this cause, unless the substances are mechanically 
mixed by stirring. 

When we come to consider the arrangement of the particles of a 
substance in dilute solution relatively to each other, we again find a 
resemblance to the gaseous state, as in the case of chlorine water, for 
instance. Water at the ordinary temperature takes up about 2*2 times 
its volume of chlorine. As the dissolved chlorine is uniformly 
* distributed in the solution, the average distance between the chlorine 
particles in the chlorine water is not very much less than the average 
distance between the particles in the chlorine gas ; and if the chlorine 
| water is only half saturated with chlorine, the distance between the 
particles is practically the same as the distance between the particles 
of the gas. So far, then, as the relative position of the particles of a 
gas and of the same substance in dilute solution is concerned, there is 


great similarity between the two states ; and as the particles of a 
gas at low pressures have very little influence on each other, so we 
may suppose that the particles of a substance in dilute solution are j 
mutually independent. If we ask what concentration of a solution 
is comparable to the concentration of a gas under ordinary condi- 
tions, we find that although the concentration is comparatively 
small, it is still such as may be frequently met with in ordinary 
laboratory work. A gram molecule of a gas at 0° and 760 mm. occupies 
22*4 litres ; a solution then containing a gram molecule of dissolved 
substance in 22*4 litres at 0° will be equally concentrated with a gas 
under standard conditions. This solution in the terminology of 
. volumetric analysis is about twenty-second normal, and solutions 
twentieth or even fiftieth normal are by no means uncommon in 
volumetric work. There is therefore an a priori probability that the 
state of a substance in dilute solution resembles in some respects the 
state of a gas, and it would not be surprising, therefore, to find that 
dissolved substances obey laws comparable to the gas laws. In the 
sequel we shall see that this is actually true. 

In order to get some idea of the effect of the solvent on a dissolved j 
substance, we shall consider briefly some properties of substances which 
are easily measurable both for the substances themselves and for their 
solutions. It is evident that the properties most suited for study in 1 
this respect are those which are possessed by the dissolved substance 
alone and not by the solvent. We get a good example of such a 1 
property in the case of optically active liquids dissolved in optically •* 
inactive solvents. We can easily measure the specific rotation given 
by oil of turpentine, say, in the pure state and in various inactive 
solvents. If the solvent has no influence on the dissolved body, the 
specific rotation ought to remain the same whether the substance is in 
solution or whether it is in a state of purity. The specific rotation of 
laevorotatory oil of turpentine is 37*01°. Solutions containing 10 per 
cent of this substance dissolved in various solvents gave the following l 


specific rotations, as calculated by the formula [a] = — , where p is the 

number of grams of active substance in v cubic centimetres of solution 
(see p. 139):— 

Solvent. Specific Rotation. 
Alcohol 38-49° 

Benzene 39*45° 

Acetic acid 40*22° 

There is evidently here an action of the solvent, for these numbers are 
all different from the number obtained for the pure substance, and not 
only are they thus divergent, but they differ also from each other. 
The optically active liquid alkaloid, nicotine, shows the same behaviour 
in still more striking fashion. The specific rotation of the pure 
substance is 161*5°; the specific rotation of a 15 per cent solution 




in alcohol is 141*6°, and that of a 15 per cent solution in water 
is only 75*5°. Here the specific rotation of the nicotine sinks to 
less than half its original value when the substance is dissolved in 
about six times its own weight of water, so that we cannot avoid 
the inference that the water exercises some powerful influence on the 
nicotine dissolved in it. In the other instances given above the 
solvents also play an important part in modifying the properties of 
the substances dissolved in them, although to a less extent than is 
the case with nicotine ; and it will be observed that each solvent exerts 
its own peculiar influence. The extent to which the rotatory power 





EtfryliG Tartrate 

in Ethyl Alcohol 



80 60 40 20 

Percentage of Tartrate 

Fig. 24. 

is affected by the solvent depends on the strength of the solution. 
The less of the active substance there is in the solution the more 
does its specific rotation diverge from the value for the pure substance. 
This may be seen in the accompanying diagram (Fig. 24), which gives 
the specific rotation of tartaric ether alone and in solutions of varying 
concentrations. The effect of water is again specially great, the specific 
rotation in 14 per cent aqueous solution being three times as great as 
the rotation for the pure ether. 

A curious regularity appears when we consider the rotations of 
aqueous solutions of different salts of optically active acids and bases. 
In order to obtain comparable numbers for the different salts, it is 
necessary to work, not with specific rotations, but with molecular 


rotations, i.e. the products of the specific rotation and the molecular 
weight. If we take, for example, the more soluble salts of camphoric 
acid, it appears that the value of the molecular rotation tends towards 
the same limit in each case as the dilution increases. This is clearly 
shown in Fig. 25, where the percentage strength of the solution is 
plotted on the horizontal axis, and molecular rotations on the vertical 
axis. The values for the molecular rotations are obtained by multiply- 
ing the specific rotations by the molecular weights, and dividing by 
100 in order to get convenient numbers. The curves, which are wide 
apart at the higher concentrations, come closer and closer together as 







Camphor ates 
P in Water 

50 40 30 20 10 

Percentage of Camphorate 
Fig. 25. 

the strength of solution diminishes, and might ultimately meet in the 
same point at zero concentration. This final concurrence, however, 
cannot be established directly owing to the difficulty of obtaining 
readings of sufficient exactness with very dilute solutions. 1 The same 
regularity appears with the soluble salts of other optically active acids, 
such as malic acid, tartaric acid, and quinic acid ; and also with the 
soluble salts of optically active bases, such as the cinchona alkaloids, 
quinine, cinchonine, etc. We may say then, in general, that the 
molecular rotation of the salts of an optically active acid or base always 

1 The salts of a-bromosulphocamphoric acid have a very high rotatory power, and 
it is therefore possible to investigate them in centinormal aqueous solutions. At this 
dilution the molecular rotations of the soluble salts are identical. 


tends to a definite limiting value as the concentration of the solution 
diminishes. This regularity is known as Oudemans 5 Law, and we may 
now attempt its interpretation. 

Taking the camphorates as our example, we note in the first 
place that the activity of the salts is due to the acid — the bases, 
potash, soda, etc., with which the acid is combined, being optically 
inactive. Indeed, it is only when we have such a combination 
of active acid with inactive base, or active base with inactive 
acid, that Oudemans' Law holds good. Now the salts of camphoric 
acid, to judge from the general run of the curves in Fig. 25, would 
in very strong solution have quite divergent molecular rotations ; and 
this we know definitely to be the case with the salts of malic acid, 
some of which in strong solution have a positive and some a negative 
rotation, although their rotations at extreme dilution are all negative. 
Where the influence of the solvent is least, therefore, the salts have 
different molecular rotations; where the influence of the solvent is 
greatest, they have nearly the same molecular rotation. The water 
therefore tends to bring the acidic, or active, portion of the salts in 
some way into the same state, for from the similarity of rotation we 
must argue similarity of condition, as this is no chance agreement, but 
a general law. This might come about in several ways, but here we 
shall consider only one possibility. Suppose that the water decomposed 
the salts into free acid and free base progressively as more and more 
water was added to the solution. In the strong solutions only a 
relatively small proportion of the gram-molecular weight of the salt 
will be decomposed, so that the total molecular rotation will be made 
up of a relatively small rotation for the free acid, and a relatively 
large rotation for the acid still bound up with the base in the form 
of salt. As the different salts have different rotations, their strong 
solutions will differ considerably in their molecular rotations, as the 
total rotation is here due mostly to the salt. It is quite different with 
dilute solutions in which, on our assumption, the salt is almost entirely 
decomposed into free acid and free base. Here the total rotation is 
due almost wholly to the acid in each case, the undecomposed salt 
contributing very little to the total ; so that no matter what the salt is, 
the value of the molecular rotation will be the same. The assumption, 
then, of the progressive decomposition of the salt into acid and base 
under the influence of the solvent water would thus account for the 
phenomena of molecular rotation in dilute solutions ; but it is on other 
grounds an improbable one, and fails to explain the numerical value 
of the limiting molecular rotation for the following reason : A 
consideration of the curves of Fig. 25 shows that if they intersect on 
the line of zero concentration at all, it will be at a value of about 39°. 
This value, therefore, would be on the above assumption the molecular 
rotation of camphoric acid in dilute solution ; but the value actually 
found for a solution of camphoric acid of 0*6 per cent strength is 93°, 


an altogether different number. We must therefore conclude that the 
supposition of a decomposition into acid and base by the water is 
untenable, and endeavour to find another explanation. It will be seen 
that the essence of the above attempted explanation is the assumption 
of the independence of the inactive basic and the active acidic portion 
of the salt in dilute solution. If on any hypothesis of the action of 
the water we can suppose that the active negative part of the salt is 
removed from the influence of the inactive positive part of the salt as the 
dilution increases, we have a sufficient explanation of the constancy of the 
molecular rotation in weak solutions, and it only remains to select such a 
hypothesis as shall conform with the other properties of dilute solutions, 
and give a rational account of the numerical value of the limiting 
rotation. We shall meet with such a hypothesis in a later chapter. 

When we deal with properties which are possessed by solvent and 
dissolved substances alike, it is more difficult to ascertain definitely if 
the property of the dissolved substance has been changed by the fact 
of its having passed into solution. As a rule we find that the sum of 
the values of the property for substance and solvent is not the value 
for the solution. For example, the volume of solvent and solute is 
never exactly equal to the volume of the resulting solution, although 
in some instances, such as cane sugar and water, this is nearly the 
case. Very frequently we find that solution is accompanied by con- 
traction in volume. When water and alcohol are mixed in equal 
measures there is a contraction of about 3 per cent of the total volume. 
This shrinkage may be due to a change in the specific volume of the 
water or the alcohol, or both, so that it is by no means easy to apportion 
the total volume change between the two substances. Even when we 
take a solution and dilute it by adding more of the solvent, we generally 
find that a change of volume occurs, a contraction on dilution being 
the usual result. 

According to Isidor Traube, the following general rule regulates 
the density of aqueous solutions : If we consider a quantity of the 
given solution such that it contains a gram - molecular weight of 
the dissolved substance, measure its volume and subtract from it the 
volume of the water which was added to make the solution, we obtain 
a residue which would be the molecular volume of the dissolved 
substance had no contraction taken place. The molecular volume may 
be determined directly with the pure substance, and the difference 
between it and the residue obtained as above described is constant 
and equal to 12*4 cc. There is thus a constant contraction in the 
aqueous solution when a gram -molecular weight of the substance is 
considered. Traube attributes the contraction to the solvent. This 
conclusion cannot be considered as fully established, for according 
to Traube's rule the whole contraction takes place when the dis- 
solved substance is first brought into contact with the water, nc 
further shrinkage occurring when the solution undergoes dilution 


with more water, which is not in accordance with the results of 

There is an undoubted regularity in the density of aqueous salt 
solutions. If we consider, for example, the density of normal solutions 
of a number of salts, we find that the difference in density between a 
chloride and the corresponding bromide is constant ; that the difference 
between a chloride and the corresponding sulphate is constant ; in short, 
that the difference between corresponding salts of two acids is 
approximately constant, no matter what the base is with which the 
acids are combined. On the other hand, we find that the difference 
in the densities of equivalent solutions of corresponding salts of two 
bases is always the same, and independent of the acid with which they 
are united. Examples are given in the following table, where the 
densities are those of normal solutions : — 

NH 4 




*S0 4 

N0 3 







N0 3 



NH 4 









From a consideration of this table, it is evident that we can obtain 
the density of the normal solution of any salt by adding to the density 
of a salt chosen as standard two numbers, or moduli, one of which is 
characteristic of the base of the salt, and the other characteristic of 
the acidic portion of the salt. This regularity is known as Valson's 
Law of Moduli, Valson choosing ammonium chloride as his standard, 
because its normal solution had the smallest density of any of the 
salts he investigated. 

The moduli for the principal series of salts are given below : — 

NH 4 0-0000 CI 0*0000 

K 0-0296 Br 0'0370 

Na 0-0235 I 0-0733 

£Ba 0-0739 N0 3 0'0160 

iCa 0-0282 JS0 4 0'0200 

Pig 0-0221 

|Zn 0-0410 

iCu 0-0413 

Ag 0-1069 

If these moduli are added to the density of normal ammonium 
chloride solution, viz. 1*0153, the densities of the other normal salt 
solutions are obtained, the values holding good for 18°. Thus, if we 
wish to know the density of an equivalent normal solution of copper 
sulphate, we add to 1*0153 the modulus of copper plus the modulus 


of the sulphates, viz. 0-0413 + 0*0200, and obtain 1*0766 as the 
result, in accordance with the experimental number. 

Not only is the law of moduli valid for normal solutions, but 
also for solutions of other (moderate) concentrations, the moduli being 
multiplied by the strength of the solution expressed in terms of a 
normal solution as unity, and then added to the density of the 
ammonium chloride solution of corresponding strength. We have the 
following values for the densities of various ammonium chloride 
solutions, which can be used as standards : — 











Should we wish to know the density of a thrice normal solution of 
calcium bromide, we add to 1*0438 three times the sum of 0*0282 and 
0-0370, viz. 0-1956, and obtain 1*2394, the value actually found by 
experiment being 1*2395. 

It is found that other properties of salt solutions besides the 
density are susceptible of similar treatment by the method of moduli. 
One salt solution is taken as standard, and from the value for it the 
values of the others may be obtained by adding the modulus for the 
acidic portion and the modulus for the basic portion of the salt under 
consideration. It will be observed that, strictly speaking, the optical 
rotations of dilute salt solutions are treated by means of moduli. If 
we take an inactive salt as standard, we can get the value of the 
rotation of any salt by adding together a modulus for the acid and a 
modulus for the basic portion of the salt. Thus if we compare in the 
case of the camphorates dilute solutions of equivalent strengths, we 
find that their rotations are all practically the same, and equal to 
what we may term the modulus for the acidic portion of the salt, the 
moduli for the different inactive basic portions being all equal to zero. 

From what has been said above, it will appear already that the 
properties of a substance in general undergo an alteration when the 
substance is dissolved in a liquid. The question as to whether the 
process of solution is to be regarded as chemical or physical has been 
much discussed, and this alteration in the value of well-defined 
properties has led many chemists to the conclusion that solution must 
be classed amongst the chemical processes. This conclusion is 
supported by the existence of heat changes accompanying solution as 
they accompany all chemical actions. It must be remembered, how- 
ever, that heat changes, as well as changes of volume and other 
properties, accompany the purely physical processes of melting, 
vaporisation, and the like. There can be no reasonable doubt that the 
dissolved substance and the solvent react on each other so as to 
influence each other's properties, but at present we are without an} 


satisfactory theory as to the origin or nature of this influence, and even 
without empirical regularities, except in a few special cases, to enable 
us to say in any given instance how the influence is most likely to 
become apparent. On the other hand, if we look upon dilute 
solutions with respect to volume, pressure, and temperature relations 
as affecting the dissolved substance, we find we can neglect the 
influence of the solvent altogether, and still obtain simple laws of the 
most general applicability, as will be shown in the succeeding chapters. 



We have seen in the preceding chapter that in some respects 
there is considerable analogy between the state of a substance existing 
as gas and the state of a substance in dilute solution. In the present 
chapter it will be shown that the analogy is more than superficial, 
and that laws exist for substances in dilute solution which are quite 
comparable with the simple laws for gases. These gas laws (cp. Chap. 
IV.) connect together the volume, pressure, and temperature of the 
gaseous substances to which they apply, so that we have first to 
consider what we are to understand by the volume, pressure, and 
temperature of a substance in solution. The temperature of a substance 
in solution is evidently the temperature of the solution itself. The 
volume of a gas we take to be the volume in which the substance as 
gas is uniformly distributed. Now the volume in which a dissolved 
substance is uniformly distributed is the volume of the solution, so that 
this volume corresponds to gaseous volume. There still remains to find 
for solutions the analogue of gaseous pressure. In the case of gases, 
the pressure we consider is that on the walls of the containing vessel, 
and may be measured directly. With substances in solution it is 
different. Here the pressure on the walls of the containing vessel 
is not the pressure of the dissolved substance, but is the gravitational 
pressure of solvent and solute combined. If we could exactly 
counteract the force of gravitation, there would be no pressure of 
the liquid on the walls of the vessel at all. There is therefore for 
substances in dilute solution no obvious magnitude corresponding to 
gaseous pressure; and yet, until this magnitude was discovered, no 
progress was made in the theory of dilute solutions. 

An experiment with gases will serve to show in what direction 
the analogue of gaseous pressure might be sought. In the case of the 
liquid solution we have two substances, and we wish to estimate the 
pressure of one of them. Can we in the case of a mixture of two 
gases find a method for measuring directly not only the total pressure 
of the two gases, but the partial pressure contributed by one of them % 


There is a theoretical method by means of which we can do this, and 
experiments have been made which go far to confirm the theory. 
Suppose that of two gases, A and B, one of them, B, can pass through 
a certain diaphragm, whilst A cannot. Let the gas A be enclosed 
in a vessel made of the material through which A cannot pass, and 
let the vessel be connected with a manometer which will measure the 
pressure of the gas within it. Suppose that the original pressure 
of A within the vessel is half an atmosphere, and let the vessel and 
its contents be immersed in the gas B, whose pressure is maintained 
steadily at one atmosphere. The gas B by supposition can pass freely 
through the material of which the vessel enclosing A is constructed, 
and it will do so until its pressure inside the vessel is equal to its 
pressure outside the vessel, viz. one atmosphere. For, if there is to 
be equilibrium between the gas inside the vessel and the same gas 
outside the vessel, there must be no difference of pressure of B 
throughout the whole space which it occupies, for the gas A exerts 
no appreciable influence on B. Inside the vessel there is now a 
total pressure of one and a half atmospheres, one atmosphere being 
the partial pressure of B> and half an atmosphere being the partial 
pressure of A, which has remained constant at its original value, 
owing to the impermeability of the vessel to this gas. Outside the 
vessel we still have the pressure of one atmosphere, so that the 
internal pressure registered when equilibrium occurs is half an atmo- 
sphere greater than the pressure outside the vessel. The excess of 
pressure inside is evidently due to the gas A which cannot pass through 
the diaphragm, so that, by taking the difference in pressure on the two 
sides of the diaphragm, we obtain the partial pressure of the substance 
to which the diaphragm is impermeable. It is not an easy matter to get 
a diaphragm which is quite permeable to one gas, and quite imperme- 
able to another ; but palladium at a moderately high temperature 
fulfils the conditions fairly well. Palladium has the property of 
absorbing hydrogen at ordinary temperatures, and parting with the 
absorbed gas again when heated in a vacuum to temperatures above 
100°. It exhibits this behaviour with regard to no other gas, so that 
at temperatures of about 200° it forms a diaphragm permeable to 
hydrogen, but impermeable to gases such as nitrogen, carbon monoxide 
or carbon dioxide. Experiments have been made with one of these 
gases inside a palladium tube, and an atmosphere of hydrogen at 
known pressure outside the tube. Theoretically, one would expect 
the internal pressure to increase by an amount equal to the external 
pressure of hydrogen. This was found to be nearly but not quite the 
case, the actual increase amounting in different experiments to from 90 
to 97 per cent of the theoretical increase. Still the result is close 
enough to show that this method of measuring the pressure due to one 
substance in a mixture might be applied in other cases with success. 
Let us now deal with a liquid solution, say a solution of cane 


sugar in water. If we could procure a semipermeable diaphragm 
of the proper kind, i.e. one which would be permeable to water and 
impermeable to sugar, we might be in a position to ascertain the 
pressure in the solution due to the presence of the sugar, and find 
how it varied with the concentration of the solution, the tempera- 
ture, etc. 

Pfeffer, the plant physiologist, while working at the osmotic 
phenomena in vegetable cells, prepared various membranes which 
proved to be perfectly permeable to water, and impermeable to some 
substances dissolved by the water. Such membranes had been pre- 
viously discovered by Moritz Traube, who, however, did not give them 
such a form as to permit of accurate work being done with them. 
They are essentially precipitation membranes, and their formation can 
be easily studied on a small scale. If a solution of copper acetate is 
added to a solution of potassium ferrocyanide, a chocolate -brown 
precipitate of copper ferrocyanide is produced ; and if the two solu- 
tions are brought together very carefully, mechanical mixing being 
avoided, the precipitate assumes the form of a fine film or membrane 
separating the two, and impermeable to both the dissolved substances. 
The experiment may be carried out in the following fashion, which 
was proposed by Traube. A piece of narrow glass tubing about 6 
inches in length is left open at one end, and closed at the other by 
means of a piece of rubber tubing provided with a clip. Into this 
tube a few drops of a 2*8 per cent solution of copper acetate are 
sucked up, by compressing the rubber beneath the clip with the 
fingers and then releasing it. The tube is now lowered into a test- 
tube containing a few cubic centimetres of a 2*4 per cent solution of 
potassium ferrocyanide. If the liquid in the inner tube forms a plane 
surface at its mouth, which can be secured by a slight movement of 
the rubber tubing, the copper ferrocyanide is deposited as a fine 
transparent film which closes the opening of the tube. That diffusion 
of the dissolved salts is prevented by this membrane is evident from 
the fact that the membrane remains transparent and of excessive 
tenuity for a very considerable period, showing that the copper and 
potassium salts no longer come into contact. A substance such as 
barium chloride, which is easily recognisable in small quantities, may 
be added to one of the membrane-forming solutions, best to that which 
is to be placed in the inner tube, and it will be found that even when 
gravitation would aid the mixing, none of the barium chloride passes 
through the septum. 

Whilst these experiments show the possibility of existence of 
membranes permeable to a liquid solvent and not to certain substances 
which might be dissolved in it, they are of no use for an investi- 
gation into the pressure exercised by dissolved substances, for the' 
films are so delicate as to be ruptured by very slight pressures or 
mechanical disturbance. Pfeffer solved the problem by depositing such 


films in the pores of unglazed vessels of fine earthenware, such as those 
employed in experiments on the diffusion of gases. This he did by 
placing one of the membrane-forming solutions in the inside of the 
porous pot and the other solution outside. The two solutions 
gradually penetrating the wall from opposite directions, at last meet 
in the interior of the wall, and there deposit a semipermeable film 
across the pores in which they meet. The film, although as delicate 
in this as in the former case, is now capable of withstanding a much 
higher pressure, on account of the support which the material of the 
porous cell affords it. If the film is to be exposed to high pressures, 
| great precautions have to be taken in its preparation, so as to avoid 
I all possibility of rupture of the membrane ; but if it is merely desired 

I to show the phenomena qualitatively, it may be done simply as follows. 
- The most convenient form of porous vessel to use is that of a bulb 
1 provided with a neck into which a rubber stopper may be inserted. 
I These bulbs are used in gas diffusion experiments, and need no further 
I preparation than washing and soaking for a day in running water. 
I The neck of the bulb is dried and coated inside and outside with 
|l melted paraffin wax, which is allowed to solidify. A solution of 
I copper sulphate (2*5 grams per litre) is introduced into the bulb 
1 up to a level above the bottom of the paraffin coating, and the bulb 
"is then placed in a beaker, into which is poured a solution of 
4 potassium ferrocyanide (2*1 grams per litre) until the bulb is 
3 immersed up to the neck. After standing for some hours the bulb 
1 is taken out of the solution, emptied, and rinsed with water. If now 
1 the bulb is filled with a strong solution of sugar, and placed in pure 
■ water, the water will pass into the interior through the semipermeable 
membrane. This is best seen by inserting into the neck a well- 
I fitting stopper through which passes a length of narrow glass tubing 
open at both ends. As the water passes into the interior of the bulb, 
t the solution rises in the narrow glass tube, and the pressure on the 
inner surface of the semipermeable diaphragm increases. Owing to 
the resistance the water experiences in passing through the fine pores 
of the bulb the process is a slow one, but in the course of an hour a 
rise of several inches may be noted, and in twenty-four hours the 
T solution may have risen from six to ten feet in the tube. As a rule, 
the membrane prepared in this way without any special precautions 
being taken, breaks down when the pressure exceeds ten feet of water, 
and the level of the liquid no longer rises. 

With a perfect membrane capable of resisting high pressures it 
I becomes a question when the increase of pressure within the bulb will 
come to an end. Pfeffer devoted his attention to this question, and 
attained the following results. For a given solution the pressure rises 
slowly until a certain maximum value is reached, after which the 
pressure remains constant. This maximum pressure, called by 
Pfeffer the osmotic pressure, varies with the nature of the dissolved 



substance. The following table contains the maximum pressures in 
centimetres of mercury observed for one per cent solutions of the 
undernoted substances : — 

Cane sugar \ 47*1 

Dextrine 16*6 

Potassium nitrate 178 

Potassium sulphate 193 

Gum . 7*2 

The pressure was found to be dependent on the strength of the 
solution, being very nearly proportional to the concentration of the 
solution, as may be seen from the following table for cane sugar, 
concentrations being given in percentages, and pressures in centimetres 
of mercury : — 

Concentration. Pressure. Ratio. 

1 53-5 53-5 

2 101-6 50*8 
2-74 151*8 55-4 
4 208-2 52-1 
6 307*5 ' 51-3 

For potassium nitrate the ratios are less constant : — 

Concentration. Pressure. Ratio. 

0-80 130*4 163 

1-43 218*5 153 

3-3 436-8 133 

As the concentration increases, the ratio of pressure to concentration 
here diminishes, but Pfeffer showed that this was really due to the 
membrane not being perfectly impermeable to potassium nitrate, a 
small quantity of the salt escaping, especially at the higher pressures, 
so that the proper maximum was never reached. 

Temperature also influences the maximum pressure, as the 
following results with a one per cent sugar solution serve to show : — 

Temperature. Pressure. 

6-8° 50-5 

13-2 52-1 

14*2 53-1 

22-0 54*8 

There is obviously here a regular increase of pressure with rise oi 

All these experiments were made by Pfeffer in 1877, but it was 
not until 1887 that van ? t Hoff published a complete theory of dilute 
solutions which takes them as its experimental basis. The semi- 
permeable membrane furnishes us with a means of directly measuring 
a pressure due to the presence of the dissolved substance, so that we 
may take it as the analogue in dilute solutions of gaseous pressure ir 
gases. It should be noted that this pressure does not necessarily 
depend for its existence on the presence of a semipermeable diaphragm 
but is only rendered evident and measurable by its means. 


We are now in possession of all the magnitudes necessary to 
enable us to investigate the pressure, volume, and temperature 
relations of substances in dilute solutions, and to compare the 
numerical results of the investigation with the corresponding relations 
for gases. The pressure we consider is the osmotic pressure ; the 
temperature is the temperature of the solution ; and the volume is the 
volume occupied by the solution. 

Pfeffer's results show, in the first place, that the osmotic pressure 
at constant temperature is proportional to the concentration of the 
solution, i.e. to the amount of substance in a given volume. In other 
words, the osmotic pressure of a given quantity of substance is 
inversely proportional to the volume of the solution which contains 
it. Here, then, is a law in perfect analogy to Boyle's law for gases : 
the volume varies inversely as the pressure. 

Let us now consider the effect of temperature on the osmotic 
pressure, the volume of the solution remaining constant. As we have 
seen, the osmotic pressure increases with the temperature just as gas 
pressure does. To determine the exact amount of the increase, 
Pfeffer made special experiments, with the following results : — 

Cane Stjgar. 

Temperature C. Temperature Abs. Osmotic Pressure. 

14*2 287*2 51*0 

32'0 305 54*4 (54*2) 


288 -5 

Sodium Taetrate. 


56*7 (55-8) 

Temperature C. 

Temperature Abs. 
, 309-6 

Osmotic Pressure. 
156*4 (154-9) 

13-3 286-3 90-8 

37-3 310-3 98-3 (98'4) 

Prom these figures it is evident that there is a close proportionality 
between the absolute temperature of a given solution and its osmotic 
pressure. In each pair of experiments the osmotic pressure at the 
higher temperature has been calculated from the experimental value 
of the osmotic pressure at the lower temperature, on the assumption 
that the osmotic pressure is proportional to the absolute temperature, 
and the calculated value has been placed within brackets alongside 
the pressure actually measured. The difference between the observed 
and calculated values is not greater than the error of experiment. 
Here, then, we have another law exactly comparable to the law for 
gaseous substances : If the volume is kept the same, the pressure is 
proportional to the absolute temperature (cp. p. 27, law 3). 


Combining these two laws for dilute solutions, we may now say 
that the product of the osmotic pressure and the volume is proportional 
to the absolute temperature, i.e. we may write the equation 

pv = BT 

for substances in dilute solution as well as for gases, and it only 
remains now to find how the constant It is related to the corresponding 
constant for substances in the gaseous state. On p. 29 we calculated 
the value of this constant for a gram molecule of a gas, and we shall 
now proceed to evaluate it from Pfeffer ; s data for a gram molecule of 
sugar dissolved in water. 

For a one per cent solution of cane sugar at 0° Pfeffer observed that 
the osmotic pressure was 49*3 centimetres of mercury. This corresponds 
to a pressure of 49*3 x 13*59 gram centimetres (cp. p. 3). The gram- 
molecular weight of cane sugar is 342, and consequently the volume of 
a one per cent solution containing the gram-molecular weight is 34,200 
cubic centimetres. The absolute temperature of the solution is 273, so 
that we have for the constant — 

Sm M x 13^9 x 34,200 , 88>900> 

a value practically identical with the value obtained for the gas 
constant, which in the same units we found to be 84,700. 

So far, then, as pressure, temperature, and volume relations are 
concerned, the analogy between gases and substances in dilute solution 
is complete. The identity of the constant in the two cases shows that 
the osmotic pressure of a dissolved substance is numerically equal to 
the gaseous pressure which the substance would exert were it contained 
as a gas in the same volume as is occupied by the solution. In fact, 
if we imagine that the solvent is suddenly annihilated, we should have 
the osmotic pressure on the semipermeable membrane replaced by a 
gaseous pressure of equal magnitude. This similarity between gaseous 
and dissolved substances is of the utmost importance, for it enables 
us to transfer to dissolved substances conclusions arrived at from the 
consideration of the temperature, volume, and pressure relations of 
gases. For example, it at once enables us to determine the molecular 
weights of dissolved substances from simultaneous measurements of 
the temperature, volume, and osmotic pressure of the solution, just 
as the molecular weights of gases and vapours are determined from 
similar magnitudes. The accurate measurement of osmotic pressure 
is an experimental task of the utmost difficulty, and has been attempted 
by only one or two investigators, so that molecular weights can hardly 
be determined directly in this way. There are, however, other 
magnitudes, susceptible of easy and exact measurement, which are 
known to be proportional to the osmotic pressure, and these are now 


made use of for molecular-weight determinations, as will be shown in 
a subsequent chapter. 

In osmotic pressure we can recognise the cause of diffusion of 
substances in solution. Just as in gases we have movement from 
regions of higher to regions of lower pressure, so in solutions we have 
movement from regions of higher osmotic pressure to regions of lower 
osmotic pressure. Osmotic pressure, then, we take to be the driving 
force in solutions, and if we calculate its value, we find it to be very 
considerable. Thus the osmotic pressure of a normal solution is over 
22 atmospheres (cp. p. 150), or 330 lbs. per square inch. In spite of 
this high driving power, the process of diffusion in solution is, as we 
have seen, a very slow one. It has been calculated that the force 
necessary to drive a gram of dissolved urea through water at the 
rate of 1 cm. per second is equal to forty thousand tons weight. 
The resistance, then, which the water offers to the movement of the 
dissolved substance is enormous. This we must take to be due to the 
smallness of the dissolved particles. A substance in the state of fine 
dust may take many days to settle, even in a perfectly still atmosphere, 
while the same weight of substance in the compact state would fall to 
the ground in as many seconds. The driving force in the two cases is 
the same, namely, the gravitational attraction of the earth for the given 
substance, but the resistance which the air offers to the small particles 
is incomparably greater than that offered to the compact mass. 

It should be borne in mind that the osmotic pressure in a solution 
may be regarded as always present, whether a semipermeable membrane 
renders it visible or not. The osmotic pressure in the ordinary reagent 
bottles of the laboratory is of the dimensions of 50 atmospheres. 
This pressure is of course not borne by the walls of the bottle, nor 
is it apparent at the free surface of the liquid. Where the liquid 
comes in contact with the enclosing vessel, there we find a liquid 
surface, and a consideration of the magnitude of the forces at work 
in the phenomena of surface tension leads us to believe that the 
pressure at right angles to the free surface of a liquid, and directed 
towards the interior of the liquid, is measurable in hundreds and even 
thousands of atmospheres. Osmotic pressures, then, large as they are 
in ordinary solutions, are small compared to the surface pressures in 
liquids, and their existence is consequently not evident at the free 
surface of liquids. It is only when these surface pressures are got 
rid of that we can measure osmotic pressures directly. The liquid 
solvent can easily penetrate the semipermeable membrane, so at the 
semipermeable membrane there is not in the ordinary sense a liquid 
surface, and consequently there is no surface pressure of the ordinary 
type. This continuity of the liquid through the semipermeable 
partition gives us, therefore, the opportunity of determining differences 
of internal pressure in the solution and the solvent. 

Various hypotheses have been put forward to explain the nature 


of osmotic pressure, but none of them can be accounted satisfactory. 
They are based upon more or less probable suppositions as to the 
nature of the movement of the dissolved molecules, the degree of 
attraction between them and the particles of the solvent, the capillary 
phenomena in the "pores" of the semipermeable partition, and the like. 
Our ignorance of such matters is, however, so great that no profitable 
conclusions have hitherto been arrived at. 

One thing may be said about osmotic pressure which is independent 
of any supposition as to its nature, and has indeed been already indicated 
in what has preceded. The osmotic pressure of any solution is in- 
dependent of the nature of the semipermeable membrane. Pfeffer 

tested several mem- 

A B 

Solvent z 

Fig. 26. 

branes besides mem- 
branes of copper 
ferrocyanide. For ex- 

Soluent ample, membranes of 
Prussian blue or of 
tannate of gelatine can 

be deposited in a por- 
ous cell in the manner 
described for copper 

ferrocyanide, and have a similar action with regard to water and 
substances held by it in solution. Pfeffer found that in general the 
osmotic pressure of any one solution varied with the nature of the 
membrane he employed, but this was in reality due to the membranes 
not being perfectly impermeable to the dissolved substance, so that 
the maximum pressures recorded did not correspond to the actual 
osmotic pressure, but fell short of it in a degree dependent on the 
extent to which the membrane leaked. A theoretical proof may be 
given that the nature of the membrane does not influence the value 
of the osmotic pressure provided that it is perfectly impermeable to 
the dissolved substance. Suppose two membranes to exist, one of 
which, A, generates with a given solution and solvent a higher osmotic 
pressure than the other membrane B. If the pressure" in the vessel 
containing the solution is less than the osmotic pressure, liquid will 
flow through the membrane from solvent to solution ; if it is greater 
than the osmotic pressure, liquid will flow from the solution to the 
solvent. Let the solution, solvent, and diaphragms be combined into 
one working system, as in the figure. If the solution is originally at 
a greater pressure than corresponds to the value of the osmotic pressure 
generated by B, solvent will flow out through the diaphragm, and the 
pressure inside will diminish and tend to reach this value. But as 
soon as the pressure diminishes to a value less than the osmotic 
pressure generated by A, solvent will flow through A into the cell. The 
pressure inside will still be too great for B, and solvent will therefore 
continue to flow out. There will thus be a continuous flow of solvent 


through the cell from left to right, and as the conditions are not 
changed by this transference, the flow might go on indefinitely and 
the current made use of to perform work, i.e. in this way we could 
obtain a perpetual motion (cp. Chap. XXVII. ). Since this is impossible, 
our assumption that the osmotic pressures generated by the two 
diaphragms are different must be incorrect, and we are forced to 
conclude that the osmotic pressure of a solution is independent of the 
diaphragm used in measuring it, provided that the diaphragm is 
completely impermeable to the dissolved substance. 

Although the direct measurement of osmotic pressure is surrounded 
by so many difficulties, it is often possible to tell whether a solution 
has an osmotic pressure greater than, equal to, or less than another 
solution of the same or a different substance. This may be done either 
with the aid of a precipitation membrane such as Traube employed, or 
by means of a natural semipermeable membrane. When a precipita- 
tion membrane is formed at the end of a tube as described above, the 
two solutions which form the precipitate have in general different 
osmotic pressures. But the solvent water moves through the 
membrane from the solution with less to the solution with greater 
osmotic pressure. If the solution outside the cell has the greater 
concentration it gains water, becomes more dilute in the immediate 
neighbourhood of the membrane, and rises in the external liquid owing 
to its lesser specific gravity. This is easily rendered visible by means 
of a Topler apparatus, which detects very small differences in the 
refractive power of liquids. If the external solution has a smaller 
osmotic pressure than the internal solution, water will be transferred 
inwards through the membrane, and the external solution will become 
more concentrated in the neighbourhood of the membrane, the change 
betraying itself by differences in density and refractive power as before. 
If, finally, the two solutions have equal osmotic pressures, no trans- 
ference of water takes place, and there is consequently no change in 
the density or refraction of the solutions. We have here, then, a 
method for determining when two membrane - forming solutions are 
isomotic, or isotonic, a term sometimes applied to solutions having 
the same osmotic pressure. 

A method making use of natural semipermeable membranes is the 
following. It is known that the protoplasm of vegetable cells has a 
sort of skin which serves to a certain extent as a semipermeable 
membrane, for it keeps dissolved substances in the cell sap from passing 
outwards, while it admits of the free passage of water. If the proto- 
plasm of the cell then is brought into contact with pure water or a 
solution of smaller osmotic pressure than the cell contents, water will 
pass through the skin inwards to the protoplasm. If, on the other 
hand, the cell content has a smaller osmotic pressure than the solution 
with which the protoplasm is brought into contact, water will pass 
outwards through the skin. Should the external solution finally have 


the same osmotic pressure as the solution within the protoplasmic 
skin, there will be no transference of water between the cell and the 
external solution. In the case of some cells, the passage of water to 
or from the protoplasm is easily visible on account of the apparent 
increase or diminution of the volume. Thus when a suitable vegetable 
cell is brought into contact with a solution of higher osmotic pressure 
than that of the solution within the protoplasm, the granular or 
coloured cell contents are seen to shrink away from the cell wall 
owing to the loss of water and contraction in volume which they 
experience. By diluting the external solution, it is easy to find a 
concentration which just ceases to produce this contraction ; then the 
cell contents are isotonic with this solution. In the same way, a 
solution of another substance may be found which is isotonic with the 
same cell. These two solutions are then isotonic with each other. 
This last statement is proved by experiments which show that two 
solutions which have been found to be isotonic with respect to one 
kind of cell are also isotonic with regard to other kinds of cell. In 
this again we have an indication that the nature of the membrane has 
no influence on the osmotic pressure, if only it is impermeable to the 
dissolved substance. 

In what follows it will practically always be assumed that the 
osmotic pressure of a solution is strictly proportional to its concentra- 
tion. This is by no means always an exact relation, and really holds 
good only for somewhat dilute solutions. The reason is of course not 
far to seek. As has been already stated above, many of the ordinary 
laboratory solutions have osmotic pressures of nearly 100 atmospheres. 
Now, concentration of a solution corresponds to absolute density in the 
case of a gas. Boyle's law for gases states that the pressure of a gas 
is proportional to its absolute density, or inversely proportional to its 
volume, which is the same thing. But this by no means necessarily 
holds good for pressures as high as 100 atmospheres. We cannot 
expect, then, that the corresponding law for solutions — that the osmotic 
pressure is proportional to the concentration — will be exactly true at 
similar high osmotic pressures. In general, we can expect no exact 
proportionality between osmotic pressure and concentration at strengths 
above normal, and we very often find that much more dilute solu- 
tions have to be considered in order to get the simple laws to apply 
in strictness. 

An account of the preparation of osmotic cells for exact measurements 
will be found in the following paper : — 

R. H. Adie — " On the Osmotic Pressure of Salts in Solution," Journal 
of the Chemical Society, l'xix. p. 344. 



In discussing the evaporation and solidification of solutions in 
Chapters VIII. and IX., we have met with empirical laws which find 
a theoretical basis in the conception of osmotic pressure. Such are 
the law that the vapour pressure of a solution is less than the vapour 
pressure of the pure solvent by an amount proportional to the strength 
of the solution (p. 79); that the boiling point of a solution is higher 
than the boiling point of the solvent by 
an amount proportional to the strength 
of the solution ; and that the freezing 
point of a solution is lower than the freez- / 

ing point of the solvent by an amount 
proportional to the strength of the solu- 

The more strict thermodynamical 
deduction of these relations from the gas 
laws for dilute solutions is given in Chapter 
XXVII., but in this chapter we can show, 
at least approximately, the relations of 
the various magnitudes. In the first place, 
we shall consider the connection between 
osmotic pressure and the relative lowering 
of the vapour pressure. 

In the figure (Fig. 27) B represents 
a porous bulb with a semi-permeable mem- 
brane deposited within the wall (p. 1 6 1 ). It fig. 27. 
is filled with a known solution and im- 
mersed in the pure solvent. The solvent will enter the bulb until 
the solution in the tube rises to a height where the difference 
of level of the liquids inside and outside the bulb causes a 
pressure equal to the osmotic pressure of the liquid. Let this 
equilibrium be reached in an atmosphere which consists only of the 
vapour of the solvent, and let the difference in level be represented 


by h. The temperature throughout is supposed to remain at the 
constant value T in the absolute scale. 

In the first place, we note that the pressure of vapour at the level I 
of the surface of the solution must be the same inside and outside the 
tube. If it were not, being, let us assume, greater inside than outside, 
vapour would pass from the region of higher to the region of lower 
pressure. This would lower the pressure of vapour immediately over 
the surface of the solution, and some of the solvent would therefore 
evaporate in order that the pressure would regain its former value, 
which is the value in equilibrium with the solution. The original 
value outside the tube is the value corresponding to the vapour 
pressure of the pure solvent, so that the accession of vapour from the 
inner tube would increase the pressure of vapour above its equilibrium 
value, and consequently some of the vapour would condense at the 
surface of the pure solvent. The process would be then, in short, that 
some of the liquid inside the tube would distil over and condense to 
liquid outside the tube. This would evidently increase the concentra- 
tion of the solution within the bulb, and so we should no longer have 
osmotic equilibrium between the solution and the solvent. To restore 
the osmotic equilibrium, some of the solvent would pass inwards 
through the membrane and dilute the solution to its original concen- 
tration. The liquid in the tube would then regain its original height 
and vapour pressure, and the whole process of distillation would 
recommence. On the assumption, then, that the pressure of vapour 
at I is greater inside the pressure tube than outside at the same level, 
we have a continuous circulation of solvent from the solution to the 
solvent as vapour, and from the solvent to the solution as liquid 
through the semipermeable membrane. The current thus generated 
could theoretically be used to perform work, and we should there- 
fore have a form of the perpetual motion, which is impossible. 
Similarly, if the pressure at the level I is supposed to be greater 
outside the tube than inside, we should have a continuous circulation 
of solvent in the opposite direction. The conclusion we must adopt, 
then, is that the pressure within and without the tube at the same 
level I is the same. If now / is the vapour pressure of the solvent at 
the given temperature, and /' that of the solution, the difference / -/' 
is evidently the difference of pressure between the levels at the two 
liquid surfaces, Le. at the top and bottom of the height h. This 
difference in pressure is due to the weight of the column of vapour 
between the two levels on a surface of one square centimetre, and is 
equal to the product of the height and absolute density of the column 
of vapour, i.e. to hd, if d is the density expressed in grams per 
cubic centimetre. 

Let us now consider a gram-molecular weight of this vapour. For 
it we have 

pv = RT, 


where B is the ordinary gas constant. But the density d is the 
weight divided by the volume, i.e. d = M/v, where M is the molecular 
weight of the solvent in the gaseous state. The pressure of the 
gaseous solvent is /, so that we obtain v = BT/f, and 

a ~ BT 

We have seen above that f-f = hd, or, substituting the value of 

d here found,/-/ = h • — whence 

/-/'_;, M 

Now / -/' is the lowering of the vapour pressure of the solvent, and 


—~- is therefore the proportional or relative lowering, with which 

we are alone concerned. We have thus obtained an expression for 

the relative lowering of the vapour pressure in terms of the " osmotic 

height " h, and constants for the gaseous solvent. It is now an easy 

matter to express h in terms of the osmotic pressure of the solution, 

and another constant for the solvent. 

The osmotic pressure, i.e. the excess of pressure inside the cell 

over that outside, is equal to the height of the column h into the 

absolute density of the liquid. This we may denote by s, which is 

the absolute density of the pure solvent, for if the solution considered 

is very dilute, its density will not greatly differ from the density of 

the solvent. We have, then, if p represents the osmotic pressure, 

j> = hs, or h = - . Substituting this value of h in the previous equation, 

we obtain 

^'=^. (1) 

/ F sBT K } 

Here the relative lowering is expressed in terms of the osmotic 
pressure of the solution and magnitudes referring to the solvent, which 
for constant temperature are constant. It appears, then, that the 
relative lowering of the vapour pressure of a liquid by the solution 
in it of some foreign substance is at any one temperature proportional 
to the osmotic pressure of the solution and independent of the nature 
of the dissolved substance. 

By making use of the gas laws for solutions we can eliminate 
temperature from the above expression, and put it in a simpler form. 
If we express the concentration of the solution in the form that 
n gram molecules of the solute are contained in W grams of the solvent, 
then we have for n gram molecules the equation 

pv = nBT. 


Now the volume in which these n gram molecules are contained is 
equal to W, the weight of solvent, divided by s, the density of the 
solvent, i.e. v= W /s, so that 


Substituting this value in the former equation, we obtain 

f~f_nsRT M_ 
f ~ W ' sRT 

The relative lowering is now expressed in terms of the concentration 
of the solution and the molecular weight of the solvent in the gaseous 
state, which is constant. The temperature has, according to this 
result, no influence on the value of the relative lowering, a conclusion 
which is in accordance with the experimental data. For a given 
weight of any one solvent, the relative lowering is proportional to the 
number of molecules of dissolved substance in solution. Consequently, 
if we find that for certain quantities of different substances, dissolved 
in the same weight of the same solvent, the relative lowering of the 
vapour pressure of the solvent is the same, we conclude that these 
quantities contain the same number of dissolved molecules, and thus 
we obtain a method for determining the molecular weights of substances 
in solution. The molecular weight of a dissolved substance might also 
be calculated in terms of the relative lowering of the vapour pressure 
by finding the numerical value of p from equation (1), and introducing 
the osmotic pressure thus obtained into the gas equation. 

For a given amount of the same substance dissolved in the same 
weight of different solvents, we find from equation (2) that the 
relative lowering is proportional to the molecular weight of the solvent 
in the gaseous state, provided that the molecular weight of the dis- 
solved substance remains the same in the different solvents. 

The expression for the relative lowering receives a still simpler 
form if, instead of the actual weight of the solvent, we introduce 
the number of gram molecules of the solvent. The weight of solvent 
may be expressed as the product of the number of molecules into the 
gram -molecular weight of the substance in the gaseous state, i.e. as 
MN, if N represents the number of gram molecules of the solvent as 
gas. We have therefore 

f MN ' 
The relative lowering is here given as the ratio of the number of 


dissolved molecules to the number of molecules which the solvent 
would produce if it were converted into vapour. 

It must be emphasised that the number of molecules N in the 
above equation does not denote the number of liquid molecules 
in the solvent, but only the number of gaseous molecules derivable 
from the liquid. This caution is necessary, because it has frequently 
been supposed that the equation enables us to determine the molecular 
weight of the liquid solvent, which is not the case. As we shall see, 
methods exist for determining the molecular weights of liquids, but 
this is not one of them. 

In deriving the above equations, we have assumed that the 
specific gravities of the solutions considered are the same as the 
specific gravities of the respective solvents. This assumption only 
holds good for very dilute solutions, but it gives serviceable approxi- 
mations in practical work, as the following numerical examples will 
show : — 

A solution of 2^2 g. of ethyl benzoate in 100 g . of benzene showed 
a relative lowering of vapour pressure equal to , 0123, i.e. if the 
vapour pressure of pure benzene is 1, the vapour pressure of the solu- 
tion is 1 - 0'0123. The temperature at which the determination was 
made was 80° C, and at this temperature the density of benzene is 
0*812. The molecular weight of gaseous benzene is 78. If we sub- 
stitute these values in equation (1), we obtain 

^' 0*812 x 84,700 x (273 + 80)' 

whence^? = 3830 g. per square centimetre, or over 3*7 atmospheres. 

If we wish to calculate the molecular weight of ethyl benzoate 
from these data by means of equation (2), we get by substitution 

°-°^ 2 - 3 = iS) 78 ' 

whence n = 0*0158, i.e. in 2'47 g. of benzoate of ethyl there are 0*0158 
gram molecules. There is therefore one gram molecule of dissolved 
ethyl benzoate in 2'47/0'0158 = 156 g., or 156 is the molecular weight 
of ethyl benzoate when it is dissolved in benzene. It must be remembered 
that this number is only approximate, but still it is sufficient to show 
that the molecular weight of the dissolved substance is practically that 
of the gaseous substance, which is 150. Equation (3) leads to the same 
result, for in order to get the number of molecules ,N we have to 
divide the number of grams taken, viz. 100, by the molecular weight, 
viz. 78. 

The determination of the lowering of vapour pressure is somewhat 
too difficult and tedious to be of much use in fixing the molecular 
weights of dissolved substances, and it is therefore preferable to 
ascertain in its stead the elevation of the boiling point, which 






*' / 




yS Oj 

a 2 

a 2 

Fig. 28. 

is for dilute solutions nearly proportional to it, and susceptible of easy 
and rapid determination. As a solution of a non-volatile substance at 

a given temperature has a 
lower vapour pressure than 
the pure solvent, it is evi- 
dent that in order that the 
solution and solvent may 
have the same vapour pres- 
sure, e.g. equal to the 
standard atmospheric pres- 
sure, it is necessary to heat 
the solution to a higher 
temperature than the sol- 
vent. The boiling point of 
the solution is therefore 
always higher than the boil- 
ing point of the solvent. 
A consideration of the accom- 
panying diagram (Fig. 28) 
will show that for small changes, the lowering of the vapour pressure 
and the elevation of the boiling point are nearly proportional. 

In the figure the three curved lines represent the vapour pressure 
curves of a pure solvent and of two solutions of different concentrations. 
At the temperature t, the intercept aa 1 represents the actual lowering 
of the vapour pressure of the solution 1, and the ratio aa x : ta the 
relative lowering. For the temperature r we have the corresponding 
magnitudes ao^ and aa x : ra. Since for any one solution the relative 
lowering is independent of the temperature, we have aa 1 : ta = aa x : Ta. 
Similarly for the second solution aa 2 : ta = aa 2 : Ta, so that aa 1 : aa 2 = 
aa^ : aa 2 , or aa Y : a x a 2 = aa 2 : <x x a 2 . If the curves were straight lines, 
they would in virtue of these proportions meet in one point, and the 
intercepts of any straight line cutting them would always bear the 
same proportions to each other. Consider the line ab 2 parallel to the 
temperature axis. This is a line of constant vapour pressure, and cuts 
the curves at points corresponding to the temperatures t Y and t 2 . The 
intercepts a\ and ab 2 represent the elevations of the boiling point, if 
the boiling point at the atmospheric pressure is t. Now if the curves 
were straight lines we should have aa 1 : aa 2 = a\ : ab 2 > i.e. the elevation 
of the boiling point would be proportional to the lowering of the 
vapour pressure. But for small elevations, that is, for dilute solutions, 
the curves may be treated as straight lines, so that we have a pro- 
portionality between the elevation of the boiling point and the lowering 
of the vapour pressure, and thus between the elevation and the 
osmotic pressure. 

Another easily determinable magnitude which is proportional to the 
osmotic pressure is the depression of the freezing point in dilute 




solutions. By means of a diagram similar to the above we can show 
that this depression is approximately proportional to the lowering of 
the vapour pressure, and thus indirectly establish the connection with 
osmotic pressure. In 
the figure (Fig. 29) aO 
represents the vapour 
pressure curve of the 
liquid solvent, say water, 
and a§' that of the solid 
solvent, ice. The tem- 
perature /, where these 
two curves intersect, is 
the freezing point of the 
pure solvent (cp. p. 98). 
The curves 1 and 2 re- 
present as before the 
vapour - pressure curves 
of two solutions. These 
cut the ice curve at two 
points, \ and 5 2 , the cor- 
responding temperatures t Y and t 2 representing the freezing points of the 
two solutions, i.e. the temperatures at which ice and the solutions are 
in equilibrium, and at which, therefore, they have the same vapour 
pressure. Now as before we may treat the curves 0, 1, and 2 as three 
straight lines meeting in a point if we only consider small intervals. 
We have then aa 1 : aa 2 = ab l : ab 2 = tt l : tt 2 . But tt 1 is the depression of 
the freezing point for the solution 1, and tt 2 the depression for the 
solution 2. Consequently, we have the depressions of the freezing 
point proportional to the lowerings of the vapour pressure, aa 1 and aa 2 . 
The freezing-point depressions are thus proportional in dilute solutions 
to the osmotic pressures of the solutions, and can therefore be sub- 
stituted for the latter in ascertaining molecular weights. 
It has now been shown on approximate assumptions that 

t 2 t x t 

Fig. 29. 


Elevation of boiling point = cP, 
Depression of freezing point = c'P, 

where P is the osmotic pressure, and c and c' constants. These 
constants remain in each case the same for a given solvent,. and are 
valid for all dissolved substances. They correspond in their nature 
to the constant factor on the right-hand side of equation (1) in this 
chapter. They depend on the properties of the solvent, and their 
derivation from these properties will be shown in Chapter XXVII. 



1. Gaseous Substances — Vapour Density 

When we can obtain a substance in the gaseous state, the determination 
of its molecular weight resolves itself, as we have seen in Chapter 
II., into ascertaining what weight of the vapour in grams will occupy 
22*4 litres at 0° and 760 mm., or, if we deal with smaller quantities, 
what weight in milligrams will occupy 22*4 cc. In general, we cannot 
weigh these volumes of the vapour under the standard conditions. 
In the case of water, for example, it is impossible to get a pressure 
of 760 mm. of vapour at 0°, the vapour pressure of water at that 
temperature being only a few millimetres of mercury. We can, 
however, make the actual determination under any conditions we 
please, and then reduce to the standard conditions by means of the 
gas laws. 

The practical problem to be solved, then, in vapour - density 
determinations for the purpose of finding molecular weights is to 
measure the weight, volume, temperature, and pressure of a given 
amount of substance in the gaseous state. This may be done in 
various ways, as the following short description of the principal 
methods will show. 

Dumas's Method. — In this method the weight of a known volume 
of gas or vapour is determined, a globe of known capacity being filled 
with the gas at atmospheric pressure and known temperature, sealed 
off, cooled, and weighed. The volume of the globe is ascertained by 
weighing it when empty and when filled with water. Its weight 
when filled with air minus the weight of air contained in it (which 
can be calculated from the volume and the known density of air) 
gives the weight of the empty globe. This, when subtracted from 
the weight of the globe filled with the gas under investigation, gives 
the weight of that gas which fills, the given volume at the given 
pressure and temperature. The method, when applied to the vapours 
of substances liquid at the ordinary temperature, usually assumes the 


Fig. 30. 

following form: The bulb is of 50 to 100 cc. capacity, as a rule, 

and has the shape shown in the figure (Fig. 30). Several grams of 

the liquid substance are placed in 

the bulb, which is then immersed 

in a bath of constant temperature 

about 20° higher than the boiling 

point of the liquid. The liquid in 

the bulb boils and expels the air 

with which the bulb was originally 

filled. After the vapour ceases to 

escape from the narrow neck of the 

bulb, this is sealed off near the 

end with a small blowpipe flame. 

There is then in the bulb a known 

volume of vapour at a known tem- 
perature and pressure, so that all 

that has now to be done is to ascertain the weight of the vapour. 
This method is somewhat troublesome, and is 
usually applied to substances which are only vola- 
tile with difficulty. 

Hofmann's Method. — Here instead of taking 
a known volume of vapour and measuring its weight, 
we take a known weight of substance and measure 
the volume which it occupies as vapour. The 
apparatus employed for the purpose is shown in 
Fig. 31. It consists of an inner tube about a yard 
long and half an inch in bore, which is graduated 
in cubic centimetres. This is filled with mercury, 
and inverted in a mercury trough, so that at the 
top of the tube a Toricellian vacuum is formed. 
Outside this tube is a wider tube which acts as 
a vapour jacket, the vapour of a boiling liquid 
passing in through the narrow tube d, and issuing, 
together with the condensed liquid, through the side 
tube just above the mercury. A weighed quantity 
of the liquid, the density of whose vapour is to be 
determined, is introduced into the inner tube in a 
bulb or very small stoppered bottle made for the 
purpose, and containing only about a tenth of a 
cubic centimetre. The inner tube is then heated 
by the vapour from a liquid boiling in a suitable 
vessel attached to d, the boiling being continued 
until the vapour issues freely from the lower tube 
and the level of the mercury in the inner tube 
The weighed quantity of substance has now been 

converted into vapour, and occupies the volume above the mercury at 


Fig. 81. 

no longer alters. 


the top of the graduated tube, at the boiling point of the liquid used 
for heating, and under a pressure equal to the atmospheric pressure 
minus the height of the column of mercury ab. 

The external liquid used for heating the inner tube to a constant 
temperature is usually water. This evidently will vaporise any 
liquid with a boiling point under 100°, but in reality it can be used 
for liquids with much higher boiling points, for the vaporisation takes 
place in the inner tube under reduced pressure. Thus a liquid such 
as aniline, with* a boiling point of 180°, is easily vaporised in the 
Hofmann apparatus by boiling water, if the quantity taken is such 

that the vapour only occu- 
pies a small proportion of 
the total volume of the 
graduated tube. 

Victor Meyer's 
Method. — This method is 
akin to Hofmann's inas- 
much as a weighed amount 
of liquid is vaporised, but 
it differs from the other 
methods on account of the 
gas whose volume, tempera- 
ture, and pressure are 
actually determined not 
being the vapour under in- 
vestigation, but an equal 
volume of air which has 
been displaced by the 
vapour. On account of the 
simplicity of the apparatus 
and the convenience in 
manipulation, this method is 
almost universally adopted 
in practical work where great accuracy is not required. 

The apparatus consists of a cylindrical vessel of about 100 cc. capacity 
with a long narrow neck, having a top piece furnished with two side 
tubes. One of these side tubes acts as a delivery tube, and is connected 
by means of a piece of thick-walled, narrow-bored rubber tubing with 
the gas-measuring tube g, which at the beginning of the experiment 
is filled with water. Through the other side tube a glass rod projects 
into the neck, connection being made by means of a short piece of 
rubber tube, permitting a good deal of play. A weighed quantity of 
the liquid whose vapour density is to be determined is contained in 
the small bulb 5, which is held in place by the rod t. The principal 
tube is closed at the top by a cork, and contains a little asbestos at 
the bottom, in order to protect the glass from the fall of the bulb. 

Fig. 32. 


The wide cylindrical portion and a large part of the neck are heated 
by means of a liquid boiling in an external cylinder with a bulb-shaped 
end. To perform an experiment, the level of water in the measuring 
tube is adjusted to zero by moving the reservoir n, the bulb is put into 
position, and the heating started while the tube is still open at the 
top. When the temperature of the whole apparatus has become steady, 
the tube is corked, and the level of the water in the measuring tube 
is observed for some minutes. If it has altered slightly, it is re- 
adjusted to zero, and the bulb is let drop by drawing back the rod t 
for a moment. The liquid at once begins to vaporise, and air passes 
over into the measuring tube. To keep the gases in the apparatus 
at the atmospheric pressure, and thus prevent leakage, the water in the 
reservoir is kept at the same level as the water in the measuring tube 
by continually lowering the reservoir. The vaporisation is complete 
in less than a minute, as a rule, and if after two or three minutes the 
level of water in the measuring tube does not change, the volume is 
read off, the barometric pressure and the temperature on a thermo- 
meter near the measuring tube being noted at the same time. 

The distribution of temperature throughout the whole apparatus 
is the same before and after the volatilisation of the substance, and 
therefore the volume of air collected is the same as the volume of 
vapour formed, after reduction to the temperature and pressure at 
which the collected air is measured. The temperature is the 
temperature of the water, the pressure that of the atmosphere minus 
the vapour pressure of water ; for the vapour pressure of the water 
over which the gas is collected and the pressure of the gas itself 
together make up the total pressure, which is equal to that registered 
by the barometer. 

In the simplest form of apparatus the gas is collected in a 
graduated tube over water contained in a shallow dish, the side tube 
being in this case long, of narrow bore, and bent to the appropriate 

It will be seen that this method of vapour-density determination 
does not involve a knowledge of the temperature at which the 
vaporisation takes place, for since all gases are equally affected by 
changes of temperature, the contraction in volume of the hot air on 
cooling is the same as the contraction which the vapour itself would 
experience. We thus, instead of measuring the volume at the 
temperature of vaporisation, measure the reduced volume at the 
atmospheric temperature. The liquid used for heating should have 
in general a boiling point at least as high as that of the experimental 
substance, and the ebullition should be so brisk as to make the vapour 
condense two-thirds of the way up the outer tube. 

The mode of calculation of a molecular weight from the observed 
data for the vapour density may be seen from the following example. 
The bulb contained 0*1008 g. of chloroform, boiling point 61°, and was 


dropped into a tube heated by the vapour from boiling water. The 
air collected measured 22*0 cc, the temperature being 16*5°, and the 
height of the barometer 707 mm. Now the vapour pressure of water 
at 16 '5° is 14 mm., so that the actual pressure of the gas was 
707-14 = 693 mm. We have only now to solve the following 
proportion : If 100*8 mg. of chloroform vapour occupies 22*0 cc. at 16'5° 
and 693 mm., what number of milligrams will occupy 22*4 cc. at 0° and 
760 mm. ? — i.e. to evaluate the expression 

100-8 x 22*4 x (273 + 16-5) x 760 

The result we obtain is 119, the actual molecular weight of chloroform 
as calculated from its formula being a little over 118. 

It should be borne in mind that in the determination of molecular 
weights from vapour densities only approximate results are obtained, 
for we assume that the vapours obey the simple gas laws exactly, 
which is by no means the case when the vapour is at a temperature 
only a little removed from the boiling point of the liquid from which 
it is produced. As, however, the vapour density is used in conjunction 
with the results of analysis in fixing the accurate value of the molecular 
weight, an error amounting to 5 or even 10 per cent of the value 
is unimportant, the number obtained from the vapour density 
merely determining the choice between the simplest formula weight 
and a multiple of it. The molecular weight of chloroform in the 
above example can from the formula be only 118 or a multiple of 
118; and the vapour-density estimation shows conclusively that the 
simplest formula is here the molecular formula. 

2. Dissolved Substances — Osmotic Pressure 

Assuming the complete similarity in pressure, temperature, and 
volume relations of substances in dilute solution and of gases, we can 
evidently determine the molecular weight of a dissolved substance by 
simultaneous observations of its weight, temperature, volume, and 
osmotic pressure. 

Pfeffer found, for example, that a one per cent solution of cane 
sugar at 32° had an osmotic pressure of 544 mm. One gram, or 
1000 mg., of cane sugar here occupied approximately 10p cc. We 
have then as before the proportion : If 1000 mg. of sugar occupy 
100 cc. at 32° and 544 mm., what number of milligrams will occupy 
22-4 cc. at 0° and 760 mm. ? The answer is 

1000 x 22-4 x (273 + 30) x 760 

100x273x544 " 347 * 

The molecular weight of cane sugar calculated from the formula 
C 12 H 22 11 is 342. It is evident, then, that the molecular formula of 


cane sugar in aqueous solution is the simplest that will express the 
results of analysis. 

Were it not for the extreme difficulty of obtaining a membrane 
perfectly impermeable to the dissolved substance, this method would 
be the most suitable and the most accurate for determining molecular 
weights of substances in very dilute solution. 

3. Dissolved Substances — Lowering of Vapour Pressure 

An example of how a molecular weight of a dissolved substance 
may be estimated by this method has been given in the preceding 
chapter (p. 173). The method has little practical importance, and is 
scarcely ever employed. 

4. Dissolved Substances — Elevation of Boiling Point 

This is a practical method for determining the molecular weights 
of substances in solution, and is coming more and more into general 
use. An essential condition for its success is that the dissolved 
substance should not itself give off an appreciable amount of vapour 
at the boiling point of the solvent. It is only applicable, therefore, 
to substances of comparatively high boiling point, say over 200°, and 
cannot be employed with success for liquids such as alcohol, benzene, 
or water. Two forms of apparatus may be described, which differ 
principally in the mode of heating. 

Beckmann's Apparatus. — In this form of apparatus the solution 
is raised to its boiling point by the indirect heat from a burner. Now 
in ascertaining the boiling point of a liquid, it is customary to place 
the thermometer, not in the boiling liquid itself, but in the vapour 
coming from it. In this way superheating is avoided. The liquid 
itself may be at a temperature considerably above its true boiling 
point, but a thermometer placed in the vapour will show very little 
sign of this superheating. The plan, however, cannot be adopted in 
ascertaining the boiling point of a solution. The vapour which comes 
from the solution of a non- volatile substance is the vapour of the solvent, 
a part of which condenses to liquid on the bulb of the thermometer. 
Now the temperature at which the condensed and vaporous solvents 
are in equilibrium on the bulb of the thermometer is the boiling point 
of the solvent and not that of the solution, so that the temperature 
registered by a thermometer placed in the vapour from a boiling 
solution is the boiling point of the solvent, slightly raised perhaps 
by radiation from the hotter solution. It is necessary then to 
immerse the bulb of the thermometer directly in the boiling 
solution if the boiling point of the latter is to be determined, ie. the 
temperature at which the solution and the vapour of the solvent are 
in equilibrium. When, therefore, the source of heat is external 


and necessarily of a higher temperature than the boiling point which 
has to be measured, precautions of the most rigorous kind have to 
be adopted in order to prevent superheating of the liquid whose 
boiling point is in question. In Beckmann's apparatus this is done as 

follows. The boiling tube A (Fig. 
33) is about 2*5 cm. in diameter, 
provided with a stout platinum 
wire fused through its end, and 
filled for about 4 cm. with glass 
beads. The platinum wire is for 
the purpose of conducting the 
external heat into the solution so 
as to get the bubbles of vapour to 
form chiefly at one place and pre- 
vent super - heating. The glass 
beads are used in order to split 
up the large bubbles of vapour 
into smaller bubbles, so that more 
intimate mixture of the solution 
and the vapour of the solvent may 
be secured. The thermometer, 
which is divided into hundredths 
of a degree, and may be read to 
thousandths, is placed so that its 
bulb dips partly into the glass 
beads. A device for rendering 
such a thermometer available for 
solvents of widely different boiling 
points is described on p. 186. 

This inner vessel is surrounded 
by a vapour jacket, charged with 
about 20 cc. of the solvent, which 
is kept boiling during the experi- 
ment. This jacket reduces radiation 
towards the exterior to a minimum, and consequently only a com- 
paratively small amount of heat has to be afforded to the solution 
in order to make it boil, whereby the risk of superheating is greatly 
reduced. Both vessels are provided with reflux condensers, which 
may either be air condensers, as in the figure, or water condensers 
of the ordinary type, according to the volatility of the solvent. 

The small asbestos heating chamber C has two asbestos rings, h and 
h', which protect the boiling vessel from the direct action of the flame 
of the burner, and two asbestos funnels, ss> which carry off the products 
of combustion. The heat of the burners reaches the liquid in the 
vapour jacket through the ring of wire gauze visible in section at d 
as a dotted line. 

Fia. 33. 


To perform the experiment, a weighed quantity of the solvent 
(15 or 20 g.) is brought into the boiling tube, the heating begun, 
and the thermometer read off when it has become steady, which may 
not be before an hour has elapsed. The condensed solvent should 
only drop back very slowly from the condenser, so that the boiling 
must not be hastened by using a large flame. The condenser K is 
then removed, and a weighed quantity of the experimental substance 
added, best in the form of a pastille if a solid, or from a specially 
shaped pipette if a liquid. The boiling point will now be found to 
rise, and the thermometer will after a short time again become 
stationary. The difference between the first and second readings of 
the thermometer is the elevation of the boiling point. Another 
weighed quantity of the substance may now be added, and the 
temperature of equilibrium again read off. This will give a second 
value for the molecular weight. 

Landsberger's Apparatus. — Since the boiling point of a 
solution of a non-volatile substance is the temperature at which the 
solution is in equilibrium with the vapour of the solvent, we can 
bring a solution to its boiling point by continually passing into it a 
stream of vapour from the boiling solvent. As long as the solution 
is under its boiling point, some of the vapour will condense, and its 
latent heat of condensation will go to heat the solution until finally 
the boiling point is reached, when the vapour will pass through the 
solution without further condensation if no heat is lost to the exterior. 
Here there is little risk of superheating, since the vapour which heats 
the solution is originally at a lower temperature than the solution itself ; 
so that if we surround the solution with a jacket of the vaporous solvent, 
we have all the conditions for real equilibrium, at least so far as the 
determination of molecular weights is concerned. Landsberger's 
apparatus secures these conditions in a very simple manner, and a 
slight modification of it is shown in Fig. 34. 

The apparatus consists of a flask F, a bulbed inner tube N, which 
contains the solution, and a wider tube E, which is connected with a 
Liebig's condenser C. The vapour is generated in F (which contains 
the boiling solvent), passes through the solution in N, from which it 
issues through the hole H, to form a vapour jacket between the two 
tubes, and finally passes into the condenser. The lower end of the 
delivery tube K, where the vapour passes into the solution, is perforated 
with a rose of small holes, so that the vapour is well distributed through 
the liquid. The bulb prevents portions of the liquid being projected 
through H if the boiling is vigorous. 

The boiling point of the solvent is first determined by placing 
enough of the pure solvent in N to ensure that the bulb of the 
thermometer is just covered by the liquid when equilibrium has been 
attained. This quantity usually amounts to from 5 to 7 cc. The 
parts of the apparatus are then put together, and the boiling of the 


solvent in F begun. To ensure regular ebullition, it is necessary to 
place in F a few fragments of porous tile, which must be renewed every 
time the boiling is interrupted. If the ebullition is brisk, the vapour 
heats the liquid in N to the boiling point in the course of a few minutes, 
as is evidenced by the reading of the thermometer rapidly becoming 
constant. The boiling is now interrupted, the tube emptied, and the 
whole process repeated, with the addition of a weighed quantity of 

Fig. 34. 

the substance under investigation to 5-7 cc. of solvent in N. The 
difference between the temperature now observed and the former 
temperature is the elevation of the boiling point, and it only remains 
to determine the weight of solvent employed. This is done by 
detaching the inner tube N, with thermometer and delivery tube, 
and weighing to centigrams. If from this weight we subtract the 
weight of the substance taken, and the tare of the tubes, etc., we 
obtain the weight of the solvent present when the temperature of 
equilibrium was reached. 


If great accuracy is not desired, several successive determinations 
with the same quantity of substance may be made by replacing the 
tube with its charge and continuing the passage of vapour, interrupting 
the boiling from time to time to ascertain the amount of solvent 
present at the moment of reading the temperature. In this case, 
instead of determining the weight of solution, it is more convenient 
to read off its volume in cubic centimetres by having the tube N 
appropriately graduated, as shown in the figure, and removing the 
thermometer and delivery tube at each interruption in order to read 
the volume. The successive determinations are less accurate on account 
of the boiling points being observed under somewhat different con- 
ditions, the pressure of the column of solution increasing, for example, 
as the solvent condenses in N. For ordinary rough laboratory 
work a thermometer graduated into fifths of a degree is sufficiently 

The calculation of the molecular weight is carried out as follows. 
For each solvent we have a constant, which is the elevation produced 
if a gram-molecular weight of any substance were dissolved in a gram 
of the solvent. Of course such an elevation is purely fictitious as it 
stands, but it has a real physical meaning if we take it to be a thousand 
times the elevation which would be produced if a gram-molecular weight 
of the substance were dissolved in 1000 g. of the solvent. The 
hundredth part of this constant, i.e. the elevation caused by dissolving 
1 g. molecular weight in 100 g. of solvent, is often spoken of as the 
molecular elevation. For the solvents ordinarily employed the 
constants are as follows : — 






















The constants in the first column refer to 1 g. of solvent, the constants 
in the second column to 1 cc. of solvent at its own boiling point, which 
are useful if we measure the volume of the solution instead of ascertain- 
ing its weight. 

In the calculation we assume exact proportionality between the 
concentration of the solution and the elevation of the boiling point. 
We thus obtain for the molecular weight the expression 


where A is the elevation, s the weight of substance, and L the weight 
of solvent, both expressed in grams, or 


nr S k' 

where V is the volume of solution in cubic centimetres. 

As an example of the calculation we may take the elevation 
produced by camphor in acetone. An elevation of 1*09° was produced 
by 0*674 g. camphor dissolved in 6*81 g. acetone. We have, therefore, 

M ~ 6-81x1-09 - 15L 

The molecular weight of camphor, according to the formula C 10 H 16 O, 
is 152. An estimation by volume resulted as follows. An elevation 
of 1*47° was found for 8*1 cc. of an acetone solution containing 0*829 g. 
camphor. This gives 

.. 0*829x2220 1K . 
8*1 x 1*47 

The molecular weight of camphor in acetone solution is thus in 
accordance with the simplest formula that expresses its composition. 

5. Dissolved Substances — Depression of Freezing Point — Baoulfs Method 

The apparatus chiefly used for determining molecular weights by 
the freezing-point, or cryoscopic, method is that devised by Beckmann, 
and figured in the accompanying illustration (Fig. 35). It consists of 
a stout test-tube A, provided with a side tube, and sunk into a wider 
test-tube B, so as to be surrounded by an air space. The whole is 
fixed in the cover of a strong glass cylinder, which is filled with a 
substance at a temperature of several degrees below the freezing point 
of the solvent. The inner tube is closed by a cork, through which pass 
a stirrer and a thermometer of the Beckmann type. This thermometer 
has a scale comprising 6° and divided into hundredths of a degree, but 
the quantity of mercury in the bulb can be varied by means of the 
mercury in the small reservoir at the top of the scale, and thus the 
instrument can be adjusted for use with solvents having widely different 
freezing points. 

To perform an experiment, a weighed quantity (15 to 20 g.) of the 
solvent is placed in A, and the external bath is regulated to a few 
degrees below the freezing point of the solvent. Thus if the solvent 
is water, a freezing mixture at about - 5° should be placed in the ex- 
ternal cylinder C. The temperature of A is lowered by taking it out 
of the air jacket and immersing it in the freezing mixture directly 
until a little ice appears. It is then replaced in the air jacket, and the 
liquid in it is stirred vigorously. As there is invariably overcooling 
before the ice and water are thoroughly mixed, the thermometer rises 


during the stirring until it reaches the freezing point, after which it 
remains constant. This constant temperature is then read off. 

The tube A is now taken out of the cooling mixture, and a weighed 
quantity of the substance under investigation is introduced and dissolved 
by stirring, the ice being allowed to melt 
save a small residue. The tube is replaced 
in the air jacket and the temperature allowed 
to fall in order that the liquid may become 
slightly overcooled. Stirring is then recom- 
menced. The thermometer rises, remains 
constant for a very short time, and then 
slowly sinks. The maximum temperature is 
read off, and taken as the freezing point of 
the solution. The reason for the subsequent 
sinking of the temperature of equilibrium is 
plain. As long as the solution remains in 
the cooling mixture, ice continues to separ- 
ate. This results in making the remain- 
ing solution more concentrated than the 
original solution, so that its temperature 
of equilibrium with the solidified solvent 
will sink (cp. p. 63). The highest tem- 
perature registered corresponds therefore 
most closely to the freezing point of the solu- 
tion whose concentration is expressed by the 
weights of substance and solvent taken, 
although even it is evidently not high 
enough, for some of the solvent has neces- 
sarily separated out as ice before equilibrium 
can be attained at all. 

The calculation is precisely the same as 
that for the elevation of the boiling point. 
Each solvent has a constant of its own, re- 
presenting the fictitious lowering of the 
freezing point caused by dissolving one 
gram molecule of substance in one gram 
of solvent. The hundredth part of this 
constant, i.e. the depression caused by dis- 
solving 1 gram molecule of substance in 

100 grams of solvent is usually termed the molecular depression. 
The constants for the most common solvents are as follows : — 

Fig. 35. 


Acetic acid 



The formula for calculation is 

tut S & 

M= LA> 

where M is the molecular weight of the dissolved substance, s its 
weight in grams, L the weight of the solvent in grams, and A the 
observed depression. A solution of 1*458 g. acetone in 100 g. benzene 
showed a depression of 1*220 degrees, whence we have the molecular 

1*458 * 490Q _ KQ 
100x1*220 ~ 58 ' 6 * 

The formula C 3 H 6 requires 58. 

An indispensable requirement of this method is that the solvent 
should separate out in the pure state without admixture of the 
dissolved substance. If this does not occur, the method is worthless 
as a practical means of ascertaining molecular weights. 

6. Pure Liquids — Surface Tension 

A method for determining the molecular weight of pure liquids, 
as distinguished from substances in liquid solution, was indicated by 
the Hungarian physicist Eotvos in 1886, but received no attention until 
it was taken up in a practical manner by Ramsay and Shields in 1893. 

From theoretical considerations Eotvos reasoned that the expression 

y(Mv)% , 

where y is the surface tension, M the molecular weight, and v the 
specific volume, would in the case of all liquids be affected equally by 
the same change of temperature. 

According to the simple gas laws, the expression 


is affected equally by temperature for all gases. There is an obvious 
similarity between these two expressions. For the pressure p in the 
one we have the surface tension y in the other. For Mv, the molecular 
volume in the one, we have (Mv)%, the molecular surface in the other. 

In the case of gases we might calculate the molecular weight from 
the relation as follows. The ratio of the change in the expression to 
the corresponding change of temperature is 

P^P^^ whence Jf- <^^\ 

where c is a constant having the same value for all gases. If we 
determine this constant once for all in the case of one gas taken as 


standard, we can calculate the molecular weights of other gases in 
terms of the molecular weight of the standard gas. Similarly for 
liquids we have 

yoW 1 

h \ y V - yiV S 



where k is a constant having the same value for all liquids. If then 
we determine the numerical value of this constant for one standard 
liquid, we can calculate the molecular weight of other 
liquids in terms of the molecular weight of this 
standard liquid. It should be mentioned that both 
the above expressions hold good only when the 
molecular weight does not change with the tempera- 
ture, and are not applicable to gases like nitrogen 
peroxide in the one case, or liquids like water in the 
other, where there is such a change. 

The method for determining the surface ten- 
sion adopted by Ramsay and Shields was to measure 
the capillary rise of the liquid in a narrow tube. 
The simplest form of apparatus they used is shown 
in Fig. 36. FG- is the capillary tube, open at the 
top and blown out to a small bulb at the bottom, in 
which there is a minute opening to admit the liquid 
contained in the wider tube A. D is a closed 
cylinder of very thin glass, which contains a spiral of 
iron wire and is connected with the capillary by 
means of a fine glass rod E. The capillary tube 
and liquid under investigation are introduced into 
A through the tube C before it is drawn out and 
sealed. After being drawn out at I, the open end of 
C is connected with the air pump, and the liquid 
within the tube boiled under diminished pressure, 
with application of heat if necessary. While the 
vapour is still issuing from the tube the narrow 
portion is rapidly sealed off at I. The tube now con- 
tains nothing but the liquid and its vapour, and is 
ready for the experiment. In order to maintain 
the liquid at a constant temperature, the tube is 
surrounded from L upwards by a mantle through 
which flows a stream of water heated to the desired 
point. HH represents the section of a magnet which 
by its attraction for the iron spiral is made to adjust 
the level of the capillary so that the liquid within 
it is always at the same, place G-, a few millimetres from the end, 
where the bore of the capillary has been previously determined by 



means of a microscope and micrometer scale. The difference of 
level between the liquid within and without the capillary is read off 
by a telescope and scale attached to the apparatus. The temperature 
is then changed and a fresh reading made, in order to ascertain the 
temperature- variation. 

To obtain the surface tension y from the observed capillary rise, 
we have the approximate formula 

\grdlh = y, 

where g is 981, the gravitational acceleration in cm. -f sec. 2 , h the 
capillary height. in centimetres, r the radius of the capillary tube at 
G in centimetres, and d the density of the liquid at the temper- 
ature of observation. The value of y is then obtained in dynes per 

The following values were observed by Ramsay and Shields for 
carbon bisulphide : — 

Radius of capillary '0129 cm. 

Temperature 19 -4° = Z 46 'l° = h 

Capillary height 4*20 cm. 3*80 cm. 

Density 1*264 1'223 

From these numbers we have the surface tensions 

y = 0'5 x 981 x 0-0129 x 1-264 x 4'20 = 33*58 at 19*4° 
7l = 0-5 x 981 x 0-0129 x 1-223 x 3*80 = 29*41 at 46-1°. 

The mean value of k for various liquids is -2*12, so that for the 
molecular weights we have, remembering that v=l/d, 

_ c -2-12(19-4 -46-1) |l 

M== \ 33-58/l-264t-29-41/l*223t} = 8 }' 6 ' 

The molecular weight of carbon bisulphide corresponding to the 
formula CS 2 is 76. The divergence here found between the 
theoretical molecular weight and that calculated from the variation of 
the surface tension with temperature is considerable, and is connected 
with the fact that the constant h has not precisely the same value for 
all liquids, but varies nearly 20 per cent, although most liquids give 
constants not far removed from the mean value - 2*12. If we 
estimated the molecular weights of gases from the change in the 
product of pressure and molecular volume with the temperature as 
suggested above we should find similar variations, as it is only for the 
gases which are liquefied with difficulty that we have nearly the same 
temperature coefficient. Thus there is a difference of 6 per cent 
between the coefficients of expansion of hydrogen and sulphur dioxide, 
and this would lead to a corresponding error in the molecular weight 
of the one referred to that of the other if we used the method 
analogous to that used for liquids. 


As we see, the surface-tension method only gives us the molecular 
weight of one liquid as compared with that of another liquid, and does 
not in itself afford us evidence of the relation between the molecular 
weight of a substance in the liquid state compared with that of the 
same substance in the gaseous state. A discussion of this point will 
be given in the next chapter. 

7. Other Methods 

Besides the methods already mentioned, there are other methods 
of determining molecular weights of substances in solution based on 
the osmotic-pressure theory. These methods are not in general use, 
but occasionally they afford valuable information where the usual means 
are unavailable. If we take a liquid such as ether, which is only 
partially miscible with water, and determine its solubility in the water, 
we find that this solubility is diminished if we dissolve in the ether a 
substance which is at the same time insoluble in the water. Just as 
the solution of a substance in ether diminishes the vapour pressure 
of the ether, so also it diminishes its solubility in any liquid with 
which it is partially miscible. And the diminution of solubility gives 
us a means of ascertaining the molecular weight of the substance dis- 
solved in the ether just as the diminution of the vapour pressure 
does. In the case of liquids the method is of no special value, but 
in the case of "solid solutions" the results obtained are of some 

The so-called solid solutions with which we have chiefly to deal are 
isomorphous mixtures, i.e. crystals which are uniformly composed of two 
crystalline substances which present similarity in crystalline form as 
well as of chemical composition. Such mixed crystals are in a sense 
comparable with liquid solutions, and one crystalline substance may 
be said to be dissolved in the other. The method of diminished 
solubility can be applied to find the molecular weight of the dissolved 
substance. If the mixed crystal consists of a large quantity of the 
substance A> in which a small quantity of the substance B is dissolved, 
the molecular weight of jR.may be determined by finding the diminu- 
tion of the solubility of A in any solvent which it occasions. In the 
practical investigation the difficulty is encountered that both A and B 
are usually soluble in the same solvents on account of their chemical 
analogy, without which there can be no isomorphous mixture. The 
calculation in such a case becomes more complex, and the results are 
of doubtful significance. 

Methods based on density determinations (cp. p. 154) have 
been proposed for determining the molecular weights of substances in 
solution, in the liquid, and even in the solid state. These methods, 
however, seem often to fail in critical cases, and sometimes lead to 


results not in harmony with those of the other better- established 
methods, so that they have as yet not met with general acceptance. 

For a more detailed description of the methods not given in Ostwald, 
Physico-Chemical Measurements, see 

Ramsay and Shields, Journal of the Chemical Society, vol. lxiii. (1893), 
p. 1089 : " The Molecular Complexity of Liquids." 

Walker and Lumsden, ibid. vol. lxxiii. (1898), p. 502 : "Determination 
of Molecular Weights — Modification of Landsberger's Boiling Point Method." 



The molecular weights as determined by any of the methods of the 
preceding chapter are average molecular weights. As a rule, the 
molecules of any one substance under given conditions are all of 
the same size, so that in most cases the molecular weight as determined 
is perfectly definite. But even with gases we have sometimes to deal 
with molecules of a single substance which are not of the same magni- 
tude, with the result that the molecular weight determined from 
observation is not the weight of any one kind of molecule, but a 
weight intermediate between real extreme values. Thus the molecular 
weight of nitrogen peroxide deduced from its vapour density under 
atmospheric pressure at 4° is 74*8, while under atmospheric pressure 
at 98° it is 52. Now these molecular weights cannot belong to 
molecules all of one kind, for the molecular weight corresponding to 
the simplest formula N0 2 is 46, while for the next simplest, N 2 4 , it is 
double this, or 92. It is evident, therefore, that under the given 
conditions we must be dealing with a mixture of the simple and 
double molecules, and that the observed molecular weights are merely 
average molecular weights of all the molecules present. We have 
here, then, a case of a substance existing in two different states 
of molecular complexity under the same conditions. Experiment 
shows that raising the temperature or lowering the pressure 
favours the existence of the simple molecules, whilst lowering the 
temperature or raising the pressure favours the existence of the double 

The formation of complex, usually double, molecules is frequently 
encountered with vapours at temperatures near the boiling points of 
the liquids from which they are derived. The vapours of the fatty 
acids, for example, have in the neighbourhood of the boiling points of 
the liquids, molecular weights considerably above those found when 
the density of the vapour is taken at a higher temperature. It should 
be remarked, however, that this is by no means the case for all 
liquids, and is on the whole exceptional, although it is true that we 



very seldom get numbers for the vapour density exactly equal to the] 
theoretical value when the vapour is near its point of condensation. 

The thorough-going analogy between substances in the gaseous 
and dissolved states furnishes us with a means of comparing their 
molecular weights in these two conditions. As a rule, we may say 
that the molecular weight in dilute solution is the same as the mole- 
cular weight of the substance in the state of vapour. Since, however, 
even for vaporous substances we find variations in the molecular 
weight under different conditions, we cannot expect in every case that 
there should be identity of the gaseous and dissolved molecules. In 
general, we may say that substances which tend to form complex 
molecules in the gaseous state exhibit the same tendency in solutions, 
the extent to which the formation of complex molecules proceeds 
depending largely on the nature of the solvent, as well as on the 
temperature and osmotic pressure (concentration) of the solution. 

The numbers in the following table give the percentage of nitrogen 
peroxide existing as double molecules in various solvents, the concen- 
tration in each case corresponding to an osmotic pressure of about 
seven atmospheres. At this pressure, and at the temperatures given 
in the table, the gaseous substance would exist almost entirely as 
double molecules ; as a matter of fact, the substance is liquid under 
these conditions, with the molecular formula N 2 4 as judged by the 
method of Eamsay and Shields. 


Double Molecules at 20°. 

Double Molecules at 90°. 

Acetic acid 



Ethylene chloride 






Carbon bisulphide 



Silicon tetrachloride 



This question of the influence of the solvent on the molecular 
weight of the dissolved substance is one of practical importance in the 
selection of a solvent in which to determine the molecular weight of a 
given substance by means of the freezing- or boiling-point method. 
As a rule, what we wish to obtain is the smallest molecular weight, and 
it is therefore expedient to select a solvent in which the tendency to 
the formation of complex molecules is as little marked as possible. Of 
the ordinary solvents, water and alcohol are those in which the formation 
of complex molecules is least apparent ; so that the former would be 
preferably chosen for the cryoscopic method, and the last mentioned 
for the boiling-point method. Acetone and ether come next in order, 
and are suitable for determining the elevation of the boiling point ; 
acetic acid is a similar solvent for the cryoscopic method. In benzene 
and chloroform the tendency to association of the simple molecules 1 
is often considerable, so these solvents should not be used if it is 
suspected that the substance tends to form complete molecules. 

Of the substances which show a tendency to the formation of « 


complex molecules, organic bodies containing the groups hydroxyl (OH) 
and cyanogen (CN) are of the most frequent occurrence. Thus the 
alcohols and the carboxyl acids almost invariably tend to form complex 
molecules in a solvent such as benzene. The subjoined tables serve to 
show this behaviour, the freezing-point method with benzene as solvent 
being employed. In the first column is given the number of grams 
of substance dissolved in 100 g. of benzene. 

Ethyl Alcohol, C 2 H 5 (OH) = 46 Phenol, C 6 H 5 (OH) = 94 


Molecular Weight. 


Molecular Weight. 

























Acetic Acid, 

CH 3 (COOH) = 60 

Benzoic Acid, 

C 6 H 5 (COOH) = 122 


Molecular Weight. 


Molecular Weight. 



0-567 f 
















It will be noticed that except in the case of ethyl alcohol the 
molecular weight does not even in the strongest solutions rise much 
above twice the value for the simplest molecule corresponding to the 
generally accepted formula. With ethyl alcohol the formation of com- 
plex molecules proceeds much further than this, if we assume the 
simple gas laws to hold good for the solutions of the strengths investi- 

To give an idea of the behaviour of a substance which shows no 
tendency to molecular complexity, the numbers for phenetol in benzene 
solution are subjoined. This substance is derived from phenol, and 
has the formula C 6 H 5 (OC 2 H 5 ) and the molecular weight 122. The 
solvent was again benzene, and with the removal of the hydroxyl group 
the substance has lost the power to form associated molecules. 

Phenetol, 6 H 5 (OC 2 H 5 ) = 122 


Molecular Weight. 













Even in what must be accounted a very strong solution the molecular 
weight does not here greatly depart from the normal value correspond- 
ing to the formula. 


Soluble salts, strong acids, and strong bases in dilute aqueous solu- 
tion invariably exhibit too small a molecular weight, whether this is 
determined by the osmotic-pressure, vapour-pressure, freezing-point, or 
boiling-point methods. The normal molecular weight of sodium chlor- 
ide, corresponding to the formula NaCl, should be 58*5, but all the 
above methods give for its molecular weight when dissolved in water 
numbers approximating to 30, i.e. about half the normal value. Here 
we are dealing with a dissociation instead of an association of the 
simplest molecules. It is evident that unless we halve the atomic 
weights of sodium and chlorine, and write the formula na cl, where na 
and cl represent the half atomic weights, the two molecules into which 
the normal molecule of sodium chloride dissociates cannot be the same. 
The most probable assumption to make is that the atomic weights 
retain their ordinary value, and that the normal molecule splits up into 
two different molecules, Na and Cl. The average molecular weight of 
the sodium chloride in solution would then, if the dissociation were 
complete, be half the normal atomic weight corresponding to the simplest 
formula. It is preferable to make this assumption of dissociation 
into atoms rather than the assumption that our ordinary atomic 
weights are twice what they should be, because we find that in other 
cases such an assumption would not account for the molecular weight 
observed. Thus in very dilute solutions of sulphuric acid or sodium 
sulphate, the atomic weight deduced from the boiling-point or freezing- 
point methods is considerably less than half the normal molecular 
weight, so that halving the atomic weights of the constituent atoms 
would be insufficient to produce the low molecular weight actually 
observed. The assumption that the dissociation is into products of 
different kinds receives support from what we learnt of the properties 
of salt solutions. It will be remembered that very frequently the 
properties of salt solutions were such that the positive and negative 
radicals appeared to be independent of each other (cp. p. 153). Now 
if we assume that the independence is caused by the salt actually split- 
ting up into these radicals, each of which then acts, as far as osmotic 
pressure and the derived magnitudes are concerned, as a separate 
molecule, we have an explanation both of the peculiarities of the 
physical properties of aqueous salt solutions and of the low molecular 
weights exhibited by the dissolved salts. In a subsequent chapter the 
subject of dissociation in salt solutions will be treated in greater detail. 

This dissociation is not confined to aqueous solutions alone, although 
it is exhibited by them to the greatest extent, but is found also in 
alcoholic and acetone solutions. In general it may be said that liquids 
with high dielectric constants act as dissociating solvents. These 
liquids themselves have usually associated molecules, but this is not 
invariably the case. Other substances dissolved in them exhibit little 
or no tendency to molecular association, whilst acids, bases, and salts 
as a rule undergo dissociation. 


The following numbers were obtained by the boiling-point method, 
and show the abnormally small molecular weight in aqueous and 
alcoholic salt solutions: — 

Sodium Acetate in Water, Sodium Iodide in Alcohol, 

C 2 H 3 2 Ka = 82 Nal = 150 

Concentration. Molecular Weight. Concentration. Molecular Weight. 

















In the chapter on solutions we found that a substance distributed 
itself between two immiscible solvents in such a way that there was 
a constant ratio of the concentrations of the substance in the two 
solutions, depending on the solubility of the substance in the sol- 
vents separately. This constant partition coefficient is observed, 
however, only when the dissolved substance has the same molecular 
weight in both solvents. If, for example, we shake up acetic acid 
in small quantity with benzene and water, we do not get a constant 
partition coefficient of the acid between the two solvents independent 
of the quantity of acetic acid present, as we should if the molecular 
weight were the same in both solvents; but we get a ratio of 
concentrations which varies as the values of the concentrations them- 
selves change. This is due to acetic acid in benzene solution 
consisting practically of double molecules, whereas in aqueous solution 
it consists of practically single molecules. In such a case we have the 
following rule for the concentrations in the two solvents. Let the sol- 
vents be A and B, and let the concentrations of the dissolved substance 
in these two solvents be C A and C B respectively. Then if the dissolved 
substance in the solvent A has a molecular weight n times greater than 
the molecular weight of the substance when dissolved in B t the ratios 

Z/C A IC 3 or CJCJ* 
are constant. It will be seen that when the molecular weight in both 
solvents is the same, we get the partition coefficient as a particular 
case of the general rule. In the above-mentioned instance of the 
distribution of acetic acid between benzene and water, where the 
molecular weight in benzene is approximately twice that in water, 
the ratio C w 2 jC B should be approximately constant. The following 
experiments show that this is indeed the case. In the first column 
we have the concentration of the acetic acid in the benzene, in the 
second the concentration in the water, in the third the ratio of these 
concentrations, and in the fourth the expression which should be 
nearly constant : — 

Cb Cw CwICb (Pw/Cb 

0-043 0-245 57 1*40 

0-071 0-314 4*4 1*39 

0*094 0-375 4-0 1*49 

0-149 0-500 3'4 1'69 


As the third column shows, there is no constant partition co- 

It has already been pointed out that there is a great analogy 
between the partition coefficient of a substance between two solvents 
and the solubility coefficient (or, shortly, solubility) of a gas in a 
liquid according to Henry's Law, and the mere existence of this law 
is sufficient to indicate that for the substances to which it applies 
the molecular complexity in the dissolved and gaseous states is the 
same. If the molecular complexity of a substance in the gaseous 
state is different from what it is when dissolved in a given solvent, 
HeinVs^Law, that the amount of gas dissolved is proportional to the 
pressure, i.e. that the ratio of the concentrations of the substance in 
the two states is constant, no longer holds good. Thus the solubility 
of carbon dioxide in water is not exactly proportional to the pressure, 
varying by as much as a hundred per cent, between 1 atm. and 
30 atm. This is no doubt due to the formation of gaseous C 2 4 
molecules under great pressures, as may be deduced from vapour- 
density determinations. If we make allowance for this increase of 
molecular complexity of the gaseous phase, as compared with the dis- 
solved phase, on increase of pressure, we find that the theoretical 
expression, where n now varies with the pressure, gives a fair approxi- 
mation to constancy. 

When we come to compare many properties of substances in the 
liquid state with each other, we find that liquids containing the 
hydroxyl group, such as the alcohols, water, and the fatty acids, are 
quite exceptional in their behaviour. In the first place, it must be 
noted that the boiling points of such compounds are exceptionally 
high, and this alone might lead us to suspect an exceptionally 
high molecular weight in the liquid state. As a rule, we find that 
on comparing the boiling points of similar compounds, the compound 
with highest molecular weight has the highest boiling point. If we 
substitute a CH 3 group for a C 2 H 5 group we have, in accordance with 
the general rule, a fall in the boiling point, and similarly if we 
substitute H for CH 3 . But if these groups are attached to an 
oxygen atom, we find that though the substitution of CH 3 for C 2 H 6 
lowers the boiling point, the substitution of H for CH 3 raises it 
greatly. The following substances afford examples of this be- 
haviour : — 

Boiling Point. 


Ethyl methyl ether 

C 2 H 5 . . CH3 



Dimethyl ether 

CH 3 .O.CH 3 


+ 90 

Methyl alcohol 



+ 34 




Boiling Point. 


CH 3 .COOC 3 H 7 



CH 3 .COOC 2 H 5 





+ 61 




Propyl acetate 
Ethyl acetate 
Methyl acetate 

Hydrogen acetate 

Another point in which organic hydroxy! compounds differ from 
1 most other liquids is the following. If a substance obeyed the gas 
laws exactly, its density at the critical temperature and pressure could 
easily be calculated from Avogadro's Law. Now all liquids have a 
much greater critical density than the calculated value, and for most 
of them the actual critical density is 3*85 times the theoretical. With 
hydroxyl compounds, however, the factor is greater than this normal 
value, varying from 4 to 5. This points to association of simple 
molecules under the critical conditions. 

Again, the vapour-pressure curves of most liquids do not cut each 
other when tabulated on the same diagram, but the curves of the 
hydroxyl compounds often cut each other and sometimes those of the 
"normal" liquids. This exceptional behaviour once more points to 
the existence of complex molecules in the liquids, which are progres- 
sively decomposed as the temperature rises. 

As we have seen in the preceding chapter, the surface-tension 
method for determining molecular weights gives a constant approxi- 
mately equal to - 2*1 for most liquids. The constants for the following 
substances are much lower than this mean value, and vary with the 
temperature — alcohols, fatty acids, water, acetone, propionitrile, and 
nitroethane. It is impossible to calculate the molecular weight of 
such substances by means of the formula given on p. 189, for that 
formula assumes that the molecular weight remains constant through 
the range of temperature examined. By suitably altering the formula, 
however, probable values may be obtained for substances whose molecular 
weight changes with the temperature, and a few of these are exhibited 
in the subjoined table, in which t is the temperature and x the associa- 
tion factor, i.e. the number of times the molecular weight of the liquid 
is greater than that corresponding to the ordinary formula : — 
"Water Acetic Acid 



20° 2*13 



60 1*99 



100 1-86 



140 1*72 



280 1*30 

Methyl Alcohol 

Ethyl Alcohol 



t X 

- 90° 


- 90° 2-03 

+ 20 


+ 20 1*65 



100 1*39 



180 1-15 


1-75 . 

220 1-03 


In each case a diminution of x with increasing temperature is 
observable, indicating that the associated molecules decompose as the 
temperature is raised. In all the above instances, except water, the asso- 
ciation factor is greater than 2, so that we are probably dealing with 
molecules more complex than double molecules. 

A comparison of these results with those previously obtained for 
the same substances in solution shows that the tendency to association 
in the liquid state persists, with some solvents, in the dissolved state. 
It is somewhat remarkable that the molecular complexity of alcohol 
as a liquid should appear considerably less than the molecular 
complexity when it is dissolved in benzene. In view, however, of the 
uncertainty attaching to the calculation of the molecular weight from 
surface-tension results where associating liquids are concerned, we 
cannot say definitely whether the discrepancy is a real one or not. 
It might be objected that as we compare the molecular weights of 
liquids only amongst themselves by the surface-tension method, we 
have no right to compare the molecular weight of a substance in the 
liquid state with that of the same substance in the dissolved or gaseous 
state. The mere fact of the continuity of the gaseous and liquid states 
(Chap. IX.) is in itself insufficient to indicate that the molecular 
condition is the same in the two cases, but where the surface-tension 
method shows the liquid to have a molecular weight corresponding to 
a complex molecule, there we have behaviour amenable to much less 
simple laws than hold good for the bulk of liquids. "We should be 
disposed, therefore, to assume that the origin of these comparatively 
simple laws is to be sought in the molecular condition in the liquid 
and vaporous states being the same. For if the molecular conditions 
in the two states were different, even in the case of normal liquids, it 
would be difficult to explain the abnormalities shown by liquids such 
as alcohol. That the normal molecular weight in solution is identical 
with the normal molecular weight in the gaseous state is practically 
certain from the existence of Henry's Law, and from the perfect 
analogy in pressure, volume, and temperature relations exhibited by 
dissolved and gaseous substances (cp. Chap. XXVII. ). 



If we take platinum wires from the terminals of a battery and join 
their free ends by another metallic wire, we find that a current of 
electricity flows through the system without being accompanied by 
any motion of ponderable matter. If we dip the free ends of the 
wires from the terminals into water acidulated with sulphuric acid, 
we find that an electric current again flows through the system, but 
that now the passage of the current is accompanied by chemical 
phenomena and motion of matter. Oxygen appears at one wire where 
it dips into the solution, and hydrogen at the other; and if we continue 
the passage of the current, taking measures to prevent diffusion in the 
solution, we shall find that the sulphuric acid will accumulate round 
the wire at which the hydrogen is evolved. 

We distinguish, therefore, between two kinds of electrical conduc- 
tion, viz. metallic conduction, which is unaccompanied by material 
change, and electrolytic conduction, which is essentially bound up 
with movement, and usually chemical change, in matter. In this 
chapter we are concerned with electrolytic conduction and the 
accompanying phenomena, and have first to ascertain what substances 
are conductors in this sense. 

Comparatively few pure substances act as electrolytic conductors, the 
exceptions being fused salts. Fused silver chloride conducts electricity 
freely, and is itself decomposed during the process, and there is even 
a perceptible electrolytic conduction in the substance in the solid state 
at temperatures not far removed from its melting point. It was by 
the electrolysis of fused salts that many of the metals were first 
prepared. Lithium and magnesium, for example, may be easily 
obtained by the passage of an electric current through their fused 
anhydrous chlorides ; and Davy first discovered the metals of the 
alkalies by electrolysing the fused bases, potassium and sodium 
hydroxides. At present, aluminium is manufactured on the large 
scale by the electrolysis of fused aluminium oxide, and many other 
technical applications of similar processes are being developed. 


Electrolysis of pure fused substances has as yet offered little of a 
regular character adapted to theoretical treatment from the chemical 
point of view, and the subject has not hitherto been systematically 
worked out. The result is that our systematic knowledge of 
electrolysis is confined almost entirely to the second class of 
electrolytes, namely solutions, and, in particular, aqueous solutions. 

Pure water can scarcely be called an electrolyte, its conductivity 
being so very small as to necessitate refined apparatus to detect it at all. 
Dry liquid hydrochloric acid in the same way cannot be called an 
electrolyte ; yet if we bring these two substances together, the resulting 
solution of hydrochloric acid is an excellent conductor of electricity, 
and undergoes decomposition when electrolysed. The conductivity, 
therefore, is not a property of either constituent of the solution, but of 
the aqueous solution itself. It is not every solvent that acquires the 
conducting property when a substance such as hydrogen chloride is 
dissolved in it. Chloroform, for example, does not conduct electricity 
itself, neither does a chloroform solution of hydrochloric acid. The 
nature of the solvent, therefore, plays an important part in determining 
whether the resulting solution will conduct or not. If a substance is 
such that its aqueous solution is an electrolyte, then its solution in 
ethyl and methyl alcohol will also conduct electricity, but not so well 
as the aqueous solution. Acetone ranks with the alcohols in this 
respect. Ether follows next, and solutions in chloroform or benzene 
and other hydrocarbons scarcely conduct at all. Solvents, then, which 
tend to associate substances dissolved in them (cp. p. 194) do not yield 
conducting solutions, while substances with no such tendency form 
solutions which conduct to a greater or less extent. 

The conductive property does not depend only on the nature of 
the solvent, however, but also on the nature of the dissolved substance. 
In general, it may be said that the only substances which exhibit 
conductivity in solution in any marked degree are salts, acids, and 
bases, i.e. the same class of substances which conduct electrolytically at 
high temperatures in the pure state. An aqueous solution of sugar or 
alcohol, for example, does not conduct much better than water itself, 
and cannot in the ordinary sense be called an electrolyte. It should 
be noted that the conducting solution is, properly speaking, the 
electrolyte, but by a convenient transference the term is often applied 
to the dissolved substance, the solvent in such a case being usually 
understood to be water. We therefore speak of acids, bases, and salts 
as electrolytes, meaning thereby that their aqueous solutions conduct 

It is often expedient to make a distinction between electrolytes, 
half-electrolytes, and non-electrolytes. In the first class are included 
practically all salts, together with the strong acids and bases, e.g. 
hydrochloric and sulphuric acids, potassium and sodium hydroxides. 
The half-electrolytes comprise the weak acids and bases, e.g. acetic and 


benzoic acids, ammonia and hydrazine. Non-electrolytes are neutral 
substances which are not salts, e.g. sugar, alcohol, urea. There is, 
strictly speaking, no sharp line of demarcation between these classes, 
intermediate substances existing which cannot be definitely classified 
within any one set. The distinction is based on degree of conductivity, 
in which there is no sudden break; but we may say that normal 
aqueous solutions of the electrolytes conduct electricity well, those 
of half-electrolytes conduct rather poorly, and those of non-electrolytes 
very feebly, or practically not at all. Thus normal hydrochloric 
acid has a conductivity two hundred times greater than normal acetic 
acid, and this again a conductivity many hundred times greater than 
a normal aqueous solution of alcohol. It should be noted at once by 
the student that although weak acids and bases are only half-electrolytes, 
their salts are good electrolytes. Normal potassium acetate, for 
example, has a conductivity fifty times that of acetic acid ; and norma] 
ammonium chloride a conductivity more than a hundred times as 
great as the conductivity of normal ammonia. Neglect or forgetfulness 
of this relation has often led to serious error, and it should therefore 
be impressed firmly in the memory. 

When a solution of sulphuric acid in water is electrolysed, the 
electrodes being of platinum or other resistant material, oxygen, as 
we have said, comes off at one of the electrodes and hydrogen at the 
other. The electrode at which the oxygen appears is called the positive 
electrode or anode, and *is connected with the positive pole of the 
battery which generates the ^current \ that at which the hydrogen is 
evolved is termed, the negative electrode or cathode, and is con- 
nected with the negative or zinc pole of the battery. It was observed 
by Faraday that the amount of decomposition in such an electrolyte 
is proportional tcTthe amount of electricity which flows through it. 
We have here, then, a direct proportionality between quantity of 
matter and quantity of electricity. For example, each gram of 
hydrogen liberated by an electric current corresponds to the passage 
through the electrolyte of 96,500 coulombs. It is of no moment 
whether the current which liberates the hydrogen is strong or weak, 
whether much or little time is occupied in the decomposition, whether 
the sulphuric acid solution is more or less concentrated, so long as 
hydrogen alone is evolved ; the result is always the same — a given 
quantity of electricity liberates in each case the same amount of 
hydrogen. What here holds good for hydrogen also holds good for 
other elements or groups of elements. A given quantity of electricity 
passed through a solution of copper sulphate always deposits the same 
quantity of copper on the cathode. On this depends the use of the 
hydrogen or copper voltameter, by means of which a quantity of 
electricity is measured by finding the amount of hydrogen or copper 
which it has liberated. 

Not only is the quantity of hydrogen liberated by a given amount 


of electricity unaffected by the concentration, temperature, etc., of the 
electrolytic solution, it is even unaffected by the nature of the dissolved 
substance, provided that this substance is of such a kind as to permit 
of the evolution of hydrogen at all. Thus if the same current is 
passed successively through dilute solutions of sulphuric acid, hydro- 
chloric acid, and sodium sulphate, it will be found that the same 
amount of hydrogen is liberated by the current in each solution. 

If we compare the volumes of oxygen and hydrogen evolved, 
simultaneously at the anode and cathode from a solution of sulphuric 
acid, we find that if the solution is dilute and the current has been 
passed for some time before the measurement is begun, in order to 
get rid of the effect of initial subsidiary reactions, the volume of 
hydrogen is double that of the oxygen. The gases are thus liberated 
in the proportions in which they combine, i.e. in chemically equivalent 
proportions. The same thing may be observed if we take other 
solutions. The quantity of copper deposited on the cathode from a 
solution of copper sulphate is exactly equivalent to the oxygen liberated 
at the anode by the same current. An indirect consequence of this is 
that if we send the same amount of electricity through solutions of 
sulphuric acid and copper sulphate, the amount of copper deposited 
by the current in the one solution will be equivalent to the amount 
of hydrogen liberated by the same current in the other. A quantity 
of electricity equal to 96,500 coulombs will therefore deposit 31*5 g. 
of copper from the solution of a cupric salt, as this is the amount 
equivalent in these salts to 1 g. of hydrogen. 

In general, we may say that the electrochemical equivalents of 
substances are identical with their chemical equivalents, if we define 
electrochemical equivalent as the amount of substance liberated by 
the same current as liberates 1 g. of hydrogen. This relation, 
together with the proportionality established between the amount of 
electricity and the amount of chemical action, gives the most general 
expression of Faraday's Law. 

If we inquire as to what happens within the electrolytic solution 
during electrolysis, we must assume that matter travels along with 
electricity, in order to explain the changes of concentration that occur 
round the electrodes. Faraday introduced the term ion to denote 
the matter which travels in the electrolyte, and as in each solution 
matter travels towards both electrodes, the term anion was used to 
denote the matter travelling towards the anode, and cation the 
matter travelling towards the cathode. It is not an easy matter to 
determine in any given case what the ions really are, and the views 
now held are not entirely in accordance with those of Faraday. The 
following system, however, is self-consistent and involves no contra- 
diction, while it affords a satisfactory explanation of the phenomena. 
In the aqueous solution of an acid the cation is hydrogen, and the 
anion the acid radical. In the solution of a base, the cation is the 


metal or metallic radical, e.g. ammonium, NH 4 , and the anion 
hydroxyl, OH. In the solution of a salt the cation is the metal or 
metallic radical arid the anion the acid radical. The cations carry the 
positive electricity, and therefore move towards the negative electrode 
or cathode. The anions carry the negative electricity, and therefore 
move towards the positive electrode or anode. 

The quantitative phenomena of electrolysis are accounted for if we 
assume that for monobasic acids, monacid bases, and their salts each 
gram ion is charged with 96,500 coulombs of electricity, which it loses 
when it reaches the oppositely-charged electrode. Take, for example, 
a dilute aqueous solution of hydrochloric acid. The positive ion in 
this case is hydrogen, the negative ion is chlorine. The water is 
supposed to play no part in the conductivity. Each gram of hydrogen 
is charged with 96,500 coulombs of positive electricity, and moves 
towards the negative electrode. There it is discharged and becomes 
ordinary hydrogen, which is liberated at the electrode. Now while 
this is going on at the negative electrode, an equal quantity of negative 
electricity must be neutralised at the positive electrode, as the same 
current flows through the whole circuit. This quantity of negative 
electricity is supplied by the gram equivalent of the negative radical, 
viz. 35*5 g. of chlorine. The chlorine when discharged of its electricity 
does not in general appear at the positive pole as such entirely. 
If the solution of hydrochloric acid is concentrated, the bulk of the dis- 
charged chlorine is liberated, but in dilute solutions it rather attacks 
the water of the solvent, combining with the hydrogen and liberating 
an equivalent quantity of oxygen. As a rule both oxygen and chlorine 
are produced, but if both be accurately estimated they are found to 
be together equivalent to the hydrogen evolved at the negative pole. 

If the solution electrolysed is one of sodium sulphate, the positive 
ion is sodium, and the negative ion S0 4 . From the formula of sodium 
sulphate it is evident that one S0 4 ion, or sulphion, as it was called 
by Faraday, is equivalent to two sodium ions. Thus for every 23 g. 
of sodium which loses its 96,500 coulombs of positive electricity, half 
of 96 g. of sulphion will lose 96,500 coulombs of negative electricity. 
In dealing with electrolytes we shall often find it convenient to give 
the charges of the ions in the formulae. A charge of 96,500 coulombs 
of positive electricity will be indicated by a dot attached to the gram- 
symbol of the positive ion, a charge of 96,500 coulombs of negative 
electricity will be indicated by a dash attached to the gram-symbol of 
the negative ion. Thus sodium sulphate will be written Na 2 'S0 4 ", 
and sodium chloride Na'CF. 

Neither sodium nor sulphion are capable of independent existence 
in presence of water, and are therefore not obtained as products of 
the electrolysis, the products of their action on water appearing in 
their stead. The sodium acts on water with production of hydrogen 
and sodium hydroxide, according to the equation 


2Na + 2H 2 = 2NaOH + H 2 , 

while the sulphion acts on water with production of sulphuric acid and 
oxygen : — 

S0 4 + H 2 = H 2 S0 4 + 0. 

The equations represent the action on water of equivalent quanti- 
ties of the discharged ions, the amounts of hydrogen and oxygen 
formed by the action being therefore also equivalent. According to 
this view, the liquid round the anode should become acid, and the 
solution round the cathode alkaline. This can easily be shown to be 
the case, and if proper precautions be taken to prevent diffusion 
within the liquid, the quantity of sulphuric acid formed at the anode 
is found to be exactly equivalent to the quantity of caustic soda formed 
at the cathode. 

It has been supposed in what has been said above that the material 
of the electrodes is not attacked by the discharged ions, a condi- 
tion which is practically secured if the electrodes are constructed 
of platinum, or, as often occurs in practice, of gas carbon. If we 
electrolyse the solution of a silver salt, say silver nitrate, between two 
silver electrodes, we find that the positive ion is discharged and 
deposited as silver on the negative pole. At the same time, an 
equivalent quantity of the negative nitrion N0 3 ' is discharged at the 
positive silver electrode. The nitrion in this case is neither liberated 
as such, nor does it attack the water. It combines with the silver to 
form silver nitrate, so that the whole electrolytic process has here 
consisted in the transference of silver from the anode to the cathode, 
and a change in the concentration of silver nitrate round the electrodes. 
Such a process is made use of in electroplating, the anode consisting 
of silver and the cathode of the object to be silverplated. The silver 
salt is chosen of such a type as to ensure a coherent film of silver on 
the surface of the plated object, and is usually a double cyanide of 
potassium and silver. 

In the formation of hydrogen gas from the hydrogen ions of 
hydrochloric acid it is evident that we have union of the discharged 
atoms, as each molecule of hydrochloric acid can contribute only one 
atom of hydrogen. Here then there is, strictly speaking, action of the 
discharged ions on each other. This is not uncommon, and is most 
evident in the actions of the discharged negative ions of carboxylic 
acids. If we take, for instance, potassium propionate, and subject it 
to electrolysis, the positive ion K' goes to the cathode, and the 
negative ion CH 3 . CH 2 . COO' to the anode. The discharged 
potassium ion as usual acts on the water with formation of potassium 
hydroxide and evolution of hydrogen. The discharged negative 
ion acts in a variety of ways. A portion of it acts on the water with 
production of the acid and liberation of oxygen. 


2CH 3 . CH 2 . COO + H 2 = 2CH 3 . CH 2 . COOH + 0. 

Under favourable conditions, however, the discharged anions react 
with each other according to the following equations : — 

2CH 3 . CH 2 . COO = CH 3 . CH 2 . CH 2 . CH 3 + 2C0 2 , 


2CH 3 . CH 2 . COO = CH 3 . CH 2 . COOH + CH 2 : CH 2 + C0 2 , 


2CH 3 . CH 2 . COO = CH 3 . CH 2 . COO . CH 2 . CH 3 + C0 2 . 

(Ethyl propionate) 

The amount of ethyl propionate produced is not great, but in other 
cases the corresponding compound is formed in considerable quantity. 
Butane, also, is only a subsidiary product, the chief gases evolved 
being carbon dioxide and ethylene. With other acids the compounds 
corresponding to butane form the bulk of the product of the interaction 
of the anions. 

The laws concerning the migration of the ions in opposite direc- 
tions towards the electrodes were ascertained experimentally by Hittorf . 
It has been said that in a solution of silver nitrate electrolysed between 
silver electrodes the only change is a transference of silver from the 
anode to the cathode, and a change in the concentration of the silver 
salt round the two electrodes. By properly constructing the apparatus 
so that ordinary diffusion of the dissolved salt between the two 
electrodes is prevented, the exact change in concentration in the 
neighbourhood of the electrodes can be measured, and from this change 
the relative speeds of the two ions can be calculated. 

It might be thought on a superficial consideration that the anion 
and cation must move at the same rate since they are liberated in 
equivalent proportions at the opposite electrodes. In silver nitrate 
solution, for instance, there is one nitrion discharged at the anode for 
each silver ion discharged at the cathode. The equivalence of dis- 
charge would, however, be retained for any relative rate of motion of 
the two ions, as the following scheme will show. In it the positive 
ions are represented by + and the negative ions by - . P is the 
positive electrode or anode, N is the negative electrode or cathode. 
D is a porous diaphragm to prevent convection currents. To begin 
with, let there be 6 molecules on each side of the diaphragm, as 
represented in Scheme I. 

N P 

+ + + + + +\+ + + + + + 


The concentration in each compartment may thus before any current 



is passed be represented by 6. Let a current now be passed, and let 
the cations alone be capable of movement, the anions remaining in 
their original compartments. The state after 3 cations have passed 
through the partition from the anodic to the cathodic compartment is 
represented in II. 


N P 

+ + + 

4- + + + 


Each ion without a partner is supposed to be discharged and liberated, 
and it will be seen that although the negative ion has not moved at 
all, the number of liberated positive and negative ions is the same. 
The number of complete molecules in the cathodic compartment has 
not altered ; the number of complete molecules in the anodic compart- 
ment has been reduced to 3. 

Let now both ions move at the same rate, i.e. let one negative ion 
cross the diaphragm to the right for each positive ion that crosses it to 
the left. If four ions of each kind are discharged, we shall have the 
state shown in III. 



4-4-4-4-4-4-4- +j+ 4-4-4- 

Here the concentration in both compartments has been reduced to the 
same extent, namely from 6 to 4. 

Let, finally, the positive ion move at twice the rate of the negative 
ion, i.e. let one negative ion cross the partition to the right in the same 
time as two positive ions cross it to the left. After three ions of each 
kind have been discharged, we have the scheme — 


N P 

4- 4-4- 4-4-4-4-4- 


The concentration has here fallen off from 6 to 5 in the cathodic com- 
partment, and from 6 to 4 in the anodic compartment. 

It is obvious from the diagrams that the loss of concentration in 
any of the compartments is proportional to the speed of the ion 
leaving the compartment. Thus in the last example the anodic 
compartment loses cations twice as fast as the cathodic compartment 
loses anions, so that the fall of concentration round the anode is twice 
as great as the fall of concentration round the cathode in the same 




time. If both ions move at the same rate, the concentrations in the 
two compartments fall off at the same rate, as in III. We therefore 
get the ratio of the speeds of the two 
ions from the observed falls in concen- 
tration round the two electrodes as 
follows : — 

Fall round anode Speed of cation 
Fall round cathode Speed of anion * ^_ s 

It must be borne in mind that it is the 
cation which leaves the anode, and pro- 
duces the fall of concentration round that 

In actual practice the conditions are 
usually somewhat different from those 
indicated above. For example, the rela- 
tive speeds of the ions of silver nitrate 
are most conveniently determined in an 
apparatus of the form shown in Figure 
37. In this form a diaphragm is dis- 
pensed with, the construction of the 
vessel itself preventing diffusion to a 
sufficient extent. The vessel has two 
limbs connected by a short, wide tube, 
the longer limb being closed at the lower 
end by a rubber tube and clip. The 
cathode C in the short limb consists of a 
piece of silver foil connected to the 
battery wire by a piece of silver wire. 
The anode A consists of silver wire 
which is bent in the form of a flat 
spiral. The portion which forms the 
stem is protected by being enclosed in 
a narrow glass capillary throughout the 
entire length immersed in the silver 
nitrate solution. A feeble current is 
passed between the electrodes through 
the silver nitrate solution with which 
the vessel is charged, the amount of elec- 
tricity passed being registered by means 
of a voltameter or other suitable instru- 
ment. At the expiry of some hours, half 
the solution is carefully withdrawn by opening the clip at the anode, 
and the amount of silver held in solution determined by analysis, the 
original concentration of the silver nitrate solution having been 
previously ascertained. 

Fig. 37. 


In this case the concentration round the anode does not fall off 
at all ; for although silver ions leave the cathodic limb, each nitrion 
that arrives at the anode dissolves from it one ion of silver so as to form 
silver nitrate. On the supposition that the ions move at the same 
rate, the state of the solution may be represented by Scheme V., which 
is comparable with Scheme III. 

V A 

-+ + + + + + +:+ + + + + + + 

The concentration at the cathode has fallen off as before from 6 to 4, 
but the concentration at the anode has increased from 6 to 8, the 
total number of complete molecules remaining the same before and 
after the passage of the current. It is easy, however, to ascertain how 
many cations have crossed the diaphragm to the left, and thus find 
what the fall in concentration round the anode would have been had 
no silver been dissolved from the electrode. The reading of the 
voltameter tells us the quantity of electricity which has passed through 
the solution, i.e. the number of negative ions which hav6 been dis- 
charged at the anode. In the above case it is 4. Now a molecule 
of silver nitrate appears at the anode for each negative ion dis- 
charged. If no cations had left the anodic compartment, there would 
thus be an increase in the concentration equal to 4 ; but the actual 
increase is only 2 : therefore had no silver been dissolved from the 
anode there would have been a fall of concentration equal to 2, or 
2 cations have crossed the diaphragm to the left in the same time 
as 2 anions have crossed it to the right. 

In an actual experiment a current which deposited 32*2 mg. of 
silver in a silver voltameter was passed through a solution of silver 
nitrate contained in an apparatus similar to that of Fig. 37. The fall 
of concentration at the cathode corresponded to 16*8 mg. of silver as 
silver nitrate, and the rise in concentration round the anode to the 
same, since the total quantity of silver nitrate in solution necessarily 
remained unaltered. Had no silver ions migrated from the anode, 
the rise in concentration would have been 3 2 '2, so that the fall due 
to migration of the cations is 32 '2 - 16*8 = 15*4. We have therefore 

Speed of cation, Ag _ Fall round anode _15'4_ 
Speed of anion, N0 3 Fall round cathode - 16*8 ~~ 

From the speed ratios for any substance it is easy to calculate what 
Hittorf called the transport numbers of the ions of the substance. If 
only the positive ion of a substance moves, as in Scheme II. , this ion is 
responsible for the total electricity carried, the negative ion having 
no share in the transport. If both ions move, they share the transport 


between them, and as each equivalent of the ions has the same charge, 

the share of each in the transport is evidently proportional to the 

speed at which it moves. If u and v are the speeds of migration of 

the positive and negative ions respectively, represents the share 

taken by the cation in the transport, and the share taken by 

the anion. These are the transport numbers of Hittorf. It is cus- 
tomary to denote the transport number of the anion by n. The 
transport number of the cation is therefore 1 - n, since the sum of the 

u v 
two transport numbers and is equal to 1. As has been 

u + v u+v 

indicated above, the ratio of the transport numbers of the ions is the 
ratio of their speeds, so that we have 

w__l -n 

v n 

If we wish to express n in terms of the ratio of the speeds ujv = r, we 
have, from the above equation, 

1 +r 

As a numerical example, we may again take silver nitrate. For this 
salt we have 


1 + 0-917 

= 0*522. 

This may also, of course, be got directly from the fall of concentration 
round the electrodes, which are proportional to the speeds of the ions. 
We have 

From his researches on the conductivity of dilute salt solutions, 
Kohlrausch established a very simple relation connecting the transport 
numbers and the molecular conductivity of the dissolved substance. 
By the molecular conductivity of a solution is meant its specific 
conductivity in the ordinary electrical units multiplied by the 
volume of the solution in litres which contains one gram 
molecular weight of the dissolved substance. In very dilute 
salt solutions the value of the molecular conductivity is independent 
of the concentration of the solution, and Kohlrausch found that this 
constant value was for different salts additively made up of two terms, 
one depending on the positive and the other on the negative radical, 
i.e. on the positive and negative ions. On considering any one salt, 
he found that the ratio of the terms for the two ions was the ratio of 


the speeds of migration of the ions. By properly choosing the units 
it was therefore possible to state for very dilute salt solutions, which 
exhibit a molecular conductivity independent of further dilution, the 
simple relation 

fJL = u + v, 

when fjt, is the molecular conductivity, and u and v numbers propor- 
tional to the relative speeds of the positive and negative ions. 
The numbers u and v do not of course represent the actual speeds 
of the ions, if the ordinary electrical units are employed for the 
molecular conductivity ; but it is easy to calculate the absolute 
value of the velocities in any given case, and the following 
table contains the speeds of the principal univalent ions in dilute 
aqueous solution at 18°, when the difference of potential between the 
electrodes is 1 volt. The velocities of migration are given in 
centimetres per hour. 






NH 4 















C 2 H 3 2 1-04 

The movement of the ions through practically pure water is seen, 
therefore, to be a very slow one. If we calculate the force required 
to drive 1 g. of hydrogen as ion through water at the rate of 1 cm. 
per second, it is found to be equal to about 320,000 tons weight. 
On reference to p. 165, it will be found that this number is of 
the same order of magnitude as the corresponding number calculated 
from the rate of diffusion of urea by means of the osmotic-pressure 
theory, viz. 40,000 tons. 

This coincidence would lead us to suspect that the resistance 
offered to the diffusion of substances in water and to the passage of 
ions through water under the influence of electric forces is of the same 
kind, and further inquiry serves to bear out the supposition. The 
resistance is connected with the viscosity or internal friction of the 
liquid, which may be measured by the time the liquid takes to flow 
through a narrow tube under given conditions. When the fluid 
friction increases, the resistance to the passage of substances through 
the liquid increases, and the rate of diffusion and rate of ionic migra- 
tion diminish in consequence of the increased resistance which the 
particles in motion through the fluid experience. The addition of a 
small quantity of a substance such as alcohol to water increases the 
viscosity of the water. Corresponding to this increase we find that 
the rate of diffusion is less when a substance is dissolved in water 
containing a little alcohol than the rate of diffusion when water alone 
is the solvent, no matter what the dissolved substance may be. 


Similarly the speed of ions in water containing alcohol is less than 
their speed in pure water. 

Again, when the temperature of water is raised, its fluid friction 
diminishes. Corresponding to this we have increased rate of diffusion 
and ionic migration as the temperature increases. There is even a 
rough proportionality between the different magnitudes. Thus at 
the ordinary temperature the fluid friction of water diminishes at the 
rate of about 2 per cent per degree. In close accordance with this, 
both the rate of diffusion of substances dissolved in water and the 
rate of migration of ions through water increase about 2 per cent of 
their value at 15° for a degree rise in temperature. It must be 
borne in mind that these relations are only roughly approximate, but 
the student will find them useful in their application to the mechanism 
of electrolytic conduction, which will be discussed in the following 

In connection with resistance, a word must be said as to the nature 
of that offered by jellies. If we make a 5 per cent solution of gelatine 
in hot water, it will set on cooling to a firm, stiff jelly, which we 
should be inclined to classify with solids rather than liquids. The 
jelly has very great internal friction, yet it offers little more resistance 
to the passage of diffusing substances or to moving ions than pure 
water does, so that the rate of diffusion or ionic migration is practically 
the same in an aqueous jelly as in water. The water in a jelly must 
therefore be supposed to retain its properties unchanged, and practi- 
cally to remain fluid. The comparative rigidity of the jelly as a 
whole we must therefore attribute to the gelatine. The only 
reasonable conception of a jelly then that we can make, is that the 
gelatine on setting forms a sort of fine spongy network in which the 
liquid water is held immeshed by capillary forces, ie. we must 
compare the state of the water in a jelly to the state of the water 
soaked up in a sponge or the water in the interstices of a porous cell. 
The porous pots used in galvanic elements prevent the mixing of the 
different liquids by convection, but they do not greatly hinder liquid 
diffusion proper or the passage of ions with their changes from one 
compartment to the other. The porous pot with the absorbed liquid 
is rigid, but the liquid in the wall retains its fluid properties unchanged. 

The actual determination of the molecular conductivity as usually 
carried out in chemical laboratories proceeds as follows. The 
principle adopted is to determine the resistance of the given solution 
by the arrangement known as Wheatstone's bridge, shown in the 
diagram, Fig. 38. E is a resistance box, by means of which resistances 
of known value can be introduced. W is the resistance to be 
measured, that of a piece of wire, for example. G is a galvanometer, 
and B a battery to produce current. MM' is a platinum wire* of 
uniform resistance stretched along a metre scale subdivided into 
millimetres. Connection is made between the galvanometer and any 



point on this wire by means of the sliding contact C. In general, a 
current flows through the galvanometer, but in the special case when 
the resistance of R is to that of W as the resistance of a is to that of 

b, no current flows 
through dc t and the 
galvanometer exhibits 
no deflection. To de- 
termine the resistance 
of Wwe place a known 
resistance in the box 
R, and move the con- 
tact C along the pla- 
tinum wire until the 
galvanometer shows 
no deflection. We know then that 

R: W = a:b. 

But the platinum wire is of uniform resistance, so that the resistance 
of a is to the resistance of b as the length of a is to the length of b y 
and these lengths may be read off directly on the scale along which 
the wire is stretched. 

When the resistance to be measured is that of an electrolyte, it 
is generally impossible to use a galvanometer owing to the polarisa- 
tion at the electrodes when a steady current flows through the 
electrolyte. This polarisation may be avoided by the use of an 
alternating current instead of a direct current, but then the galvano- 
meter is useless to indicate the alternating current. Its place, how- 
ever, may be taken, as Kohlrausch has shown, by the telephone, 
which is silent when no current passes through it, but sounds 
when it is traversed 
by alternating cur- 
rents. The alternat- 
ing current is got from 
the secondary coil of 
a small inductorium, 
worked by the battery 
B, which is usually a 
single bichromate cell. 
The modified arrange- 
ment is shown in Fig. 
39, where S is the 
electrolytic solution, T 
the telephone, and I the induction coil. 

The vessel which contains the solution is generally of the type 
proposed by Arrhenius, and shown in Fig. 40 in natural size. The 
electrodes are made of stout platinum discs, fitting closely to the 




cylindrical vessel. A short platinum stem from each is sealed into 
a glass tube, held firmly in the ebonite cover C. The connecting wires 
are passed down the glass tubes till they make contact with the 
platinum wires by means of a drop of mercury introduced into each 
tube. This form is most suitable for very small conductivities. For 
solutions which have greater conductivity, a modified type with a 
narrower end may be employed, which has the advantage of being 
considerably cheaper on account of the 
much smaller size of the platinum elec- 
trodes used in its construction. If the 
platinum electrodes are bright, there is no 
sharp minimum of sound in the telephone 
at any position of the sliding contact. It 
is therefore necessary to cover the surface 
of the platinum with a coating of finely- 
divided platinum, and the success of the 
method largely depends on how the platin- 
isation of the electrodes is effected. If 
a fine velvety coating of platinum black 
does not entirely cover the inner surfaces 
of the electrodes, the sound minimum is 
more or less indistinct owing to the polar- 
isations of the make and break induction 
currents not exactly neutralising each 
other, and the measurements of resistance 
in consequence are more or less doubtful. 
The platinisation is carried out by electro- 
lysing a solution of platinum chloride 
(chloroplatinic acid) between the electrodes 
by means of a direct current. The best 
solution to employ is one containing 30 
parts water, 1 platinum chloride, and FlG 40 

0'008 lead acetate. The electrolysing 

current is so regulated that there is a feeble gas evolution from 
the anode and a fairly brisk evolution from the cathode. The direc- 
tion of the current is occasionally reversed, and the platinisation is 
complete when each electrode has served as cathode for about fifteen 

A large induction coil is not necessary for the success of the 
method ; in fact, a small toy coil with a high rate of alternation works 
best in the chemical laboratory. It should be some six feet distant 
from the measuring wire, and enclosed in a box so that the sound from 
the make and break of the coil itself does not interfere with the sound 
in the telephone. 

The cell containing the electrolytic solution is immersed 'in a bath 
of constant temperature, usually 25°, fluctuations of more than a tenth 


of a degree from the mean being inadmissible, owing to the great varia- 
tion of the conductivity with the temperature. 

For chemical purposes the dilutions generally are made to increase 
in powers of 2. If the substance under investigation is sufficiently 
soluble, a solution containing a gram equivalent, or a gram molecule, 
in 16 litres is first prepared, and 20 cc. of this solution introduced 
into the electrolytic cell by means of a 10 cc. pipette twice filled. 
Another 10 cc. pipette is marked by direct experiment so as to suck 
up exactly as much water as the first pipette delivers. After the first 
reading of the conductivity has been taken, 10 cc. of the solution are 
removed by the second pipette and 10 cc. of water added by means of 
the first pipette, after which the diluted solution is well mixed by 
motion of the electrodes. The solution is now twice as dilute as for- 
merly, i.e. its dilution is 32. The reading is repeated after the 
solution has attained the temperature of the bath, and the dilution 
process is gone through anew. When the dilution has reached 1024 
the process is usually stopped, except in the case of strongly-dissociated 
salts, for the conductivity caused by impurities in the distilled water 
renders the values uncertain. 



In the preceding chapter we have become acquainted with some of the 
fundamental facts of electrolysis ; in the present chapter we proceed to 
the consideration of a mechanical scheme which affords a simple mode 
of representing them, as well as many other peculiarities of electrolytic 
solutions. Before entering on the discussion of this scheme, however, 
it is necessary to draw attention to two further facts which must be 
accounted for by any satisfactory theory of electrolysis. 

In the first place, it has been ascertained by careful experiment 

Fig. 41. 

that electrolytic solutions obey Ohm's Law as strictly as do metallic 
conductors, the current being proportional to the electromotive force 
for all values of the force. A direct consequence of this is that no 
electrical energy is expended in splitting up the dissolved salt molecules 
into their constituent ions, as was first indicated by Clausius. 

In the second place, when the circuit is completed between two 
electrodes, the products of electrolysis appear simultaneously at them, 
no matter how far apart they may be. Thus if we have, as in Fig. 41, 
a narrow glass tube ER, 1 cm. in bore and 40 cm. long, connected at 
the ends with wide tubes containing the two electrodes a and c, and 
filled with sulphuric acid, the products of the electrolysis of the 



sulphuric acid will make their appearance at the electrodes as soon 
as connection is made through a powerful battery. If the anode a 
is of copper and the cathode a piece of platinum wire, the blue colour 
of copper sulphate and the bubbles of hydrogen at k will be observed 
simultaneously. Now the first hydrogen ions cannot therefore come 
from the same molecules of sulphuric acid as the first sulphions, which 
unite with the copper to form copper sulphate, for the ions, in what- 
ever way they might be supposed to move, could not traverse a 
distance of 40 cm. in a few seconds, as may be seen from the table 
of rates of migration given in the preceding chapter. 

In the schemes for the representation of the migration of the ions 
we have assumed that there is a constant change of partners going on 
as the ions travel to the opposite electrodes. This idea was introduced 
by Grotthus, who conceived that the molecules at the two electrodes 
were split up into their positive and negative constituents under the 
influence of the electric charges on the electrodes, and that the 
intermediate molecules changed partners according to a scheme like 
the following, the first action of the electric charges being to direct 
all the positive ends of the molecules towards the negative electrode 
and the negative ends towards the positive electrode : — 




®@ ®@ ®@ ®@ ®@ 










This, however, does not get over the difficulty indicated by Clausius, an 
account of whose views may be given in the words of Clerk Maxwell. 

" Clausius has pointed out that on the old theory of electrolysis, 
according to which the electromotive force was supposed to be the 
sole agent in tearing asunder the components of the molecules of the 
electrolyte, there ought to be no decomposition and no current as long 
as the electromotive force is below a certain value, but that as soon as 
it has reached this value a vigorous decomposition ought to commence, 
accompanied by a strong current. This, however, is by no means the 
case, for the current is strictly proportional to the electromotive force 
for all values of that force. 

" Clausius explains this in the following way : — According to the 
theory of molecular motion, of which he has himself been the chief 
founder, every molecule of the fluid is moving in an exceedingly 


irregular manner, being driven first one way and then another by the 
impacts of other molecules which are also in a state of agitation. 

" This molecular agitation goes on at all times independently of 
the action of electromotive force. The diffusion of one fluid through 
another is brought about by this molecular agitation, which increases 
in velocity as the temperature rises. The agitation being exceedingly 
irregular, the encounters of the molecules take place with various 
degrees of violence, and it is probable that even at low temperatures 
some of the encounters are so violent that one or both of the compound 
molecules are split up into their constituents. Each of these con- 
stituent molecules then knocks about among the rest till it meets with 
another molecule of the opposite kind, and unites with it to form a 
new molecule of the compound. In every compound, therefore, a 
certain proportion of the molecules at any instant are broken up into 
their constituent atoms. At high temperatures the proportion becomes 
so large as to produce the phenomenon of dissociation studied by 
M. Ste. Claire Deville. 

" Now Clausius supposes that it is on the constituent molecules in 
their intervals of freedom that the electromotive force acts, deflecting 
them slightly from the paths they would otherwise have followed, 
and causing the positive constituents to travel, on the whole, more in 
the positive than in the negative direction, and the negative con- 
stituents more in the negative direction than in the positive. The 
electromotive force, therefore, does not produce the disruptions and 
reunions of the molecules, but, finding these disruptions and reunions 
already going on, it influences the motion of the constituents during 
their intervals of freedom." 

The constituent molecules referred to in the above passage are, of 
course, the positive and negative radicals of the dissolved salt, i.e. the 
cation and anion of which it is assumed to be composed. At 
any one time then we have, on the hypothesis of Clausius, some pro- 
portion of the salt molecules split up into their constituent ions, which, 
with their electric charges, move towards the appropriate electrodes. 
It must be observed that this state of partial dissociation of the dis- 
solved substance is the normal condition of the liquid, and exists 
whether there is an electric current passing through the solution or 
not. All that the electric forces do is to direct the dissociated charged 
products to the electrodes and there discharge them. Nothing has 
been said as to the proportion of dissolved substance which is thus 
dissociated into ions. For the purpose of accounting for the validity 
of Ohm's Law in electrolytic solutions, any proportion, however small, 
will suffice, provided that the small quantity is always regenerated 
by the action of the molecules themselves without any interference 
of the electrical forces. In proportion as the free ions are removed 
from the solution at the electrodes, Clausius supposes them to be 
regenerated before by the collisions of the undissociated molecules, 


so that the process of conduction and electrolysis goes on. If we are 
to give the hypothesis definiteness and precision, however, we must 
take account of the relative quantities of the electrolyte in the dis- 
sociated and undissociated states. The manner of doing this was first 
pointed out by Arrhenius, and it is to his hypothesis of electrolytic 
dissociation that we must resort if we wish to explain quantitatively 
the phenomena exhibited by electrolytic solutions, whether during 
electrolysis or in their ordinary state. 

Arrhenius supposes substances which give solutions that conduct 
electricity freely to be almost entirely split up into their constituent 
ions, while substances which yield solutions of feeble conductivity are 
supposed by him to be split up only to a very small extent. In fact, 
he proposes to measure the degree of dissociation of a substance by the 
conductivity of its solutions. On his hypothesis, only those molecules 
which are split up into their constituent ions play any part in the 
conduction of electricity, the undissociated molecules remaining idle. 
It is obvious, therefore, that the conductivity of any given solution 
depends on two factors — the number of ions in the solution, and the 
rate at which these ions move. To simplify matters we will, in what 
follows, only consider univalent ions, ie. those derived from monacid 
bases, monobasic acids, and the salts which they form by mutual 
neutralisation. Every ion derived from these substances has the same 
charge of electricity, i.e. 96,500 coulombs per gram ion. Since each 
carrier of electricity has the same load, the quantity carried can depend 
only on the number of carriers and the speed at which they move. 
Now the rate at which the ions move may, as we have seen, be deter- 
mined from the work of Hittorf and Kohlrausch. It only remains, 
therefore, to find the number of ions in any given solution. 

Kohlrausch ascertained experimentally that the molecular conduc- 
tivity of a salt increases as the dilution of the solution increases, and 
that the rate of increase of conductivity gets smaller and smaller as the 
dilution gets greater, until finally the molecular conductivity remains 
constant, although the addition of water to the solution is continued. 
This may best be figured as follows. Consider a cell of practically 
infinite height and of rectangular horizontal section, two parallel sides 
of which are of platinum, and are placed at a distance of 1 cm. from 
each other. These platinum sides may be used as electrodes. Let 
there now be introduced into the cell a gram molecular weight of 
common salt (58*5 g.) dissolved in a litre of water. If the resistance 
offered to the passage of the current through the liquid is measured in 
Siemens' mercury units (p. 6), the reciprocal of the number obtained 
represents the molecular conductivity as ordinarily expressed. We 
thus obtain the molecular conductivity of sodium chloride at the dilu- 
tion 1. If we add water so as to make up the volume to 2 litres, and 
again determine the molecular conductivity, we find that the value for 
the dilution 2 is greater than before. As we add more water so as to 


increase the volume in which the gram-molecular weight of the salt 
is contained, the molecular conductivity will also increase, finally to 
reach a limit when the dilution amounts to about 10,000 litres. The 
numbers obtained by Kohlrausch for sodium chloride at 18° are given 
in the following table : — 

Dilution = v 

Mol. Cond.=/x 


1 lit. 



2 „ 



10 ,, 



20 „ 



100 „ 



500 „ 



1,000 ,, 



5,000 ,, 



10,000 „ 


50,000 „ 


100,000 „ 


From this table it will be seen that the rate of increase of the 
molecular conductivity is much greater when the dilution is small 
than when it is great. Doubling the quantity of water when the 
dilution is 1 adds more than 6 units to the conductivity ; doubling 
the quantity when the dilution is 500 only adds 1 unit to the 
conductivity. Increasing the dilution tenfold when it is already 
10,000 has no further effect, the small variations observed in the 
last three values being due to experimental error. 

The molecular conductivity, as has been said, depends only on the 
number of ions and the rate at which they move. We have therefore 
to determine to which of these causes the increase of molecular 
conductivity on dilution is due. The rate of the ions depends on the 
resistance to their motion offered by the liquid. Now at a dilution 
of 10 we have 5 8 '5 g. of salt dissolved in 10,000 g. of water. 
So far as viscosity is concerned, this is practically pure water, and 
further additions of water will have no appreciable effect in changing 
the resistance offered to the passage of the ions. We must suppose 
therefore that the rate at which the ions travel is practically unaltered 
after a dilution of about 10 is reached, so that the increase of 
conductivity with further dilution is not due to any increase of speed 
of the ions, but to an increase in their number. In the imaginary 
cell considered above we have always the same amount of salt between 
the electrodes, but evidently as we add water we obtain a greater 
number of ions. With increasing dilution the salt then must split up 
more and more into ions capable of conveying the electricity, if we 
are to account for the increase of molecular conductivity which the 
salt exhibits. When all the salt has been split up into its ions the 
increase of molecular conductivity with dilution must cease, for further 
dilution can neither increase the speed of the ions nor augment their 
number. In the case of sodium chloride the salt is entirely dissociated 
into its ions at a dilution of 10,000, and we find that salts in general 


exhibit this behaviour. The limiting value of the molecular conduc- 
tivity corresponding to complete dissociation is called the molecular 
conductivity at infinite dilution, and is usually denoted by /*«>. 

Since in dilute solutions the addition of more water does not 
affect the speed of the ions, it is obvious that in a cell such as we con- 
sidered above the conductivity of the solution is directly proportional 
to the number of ions in it. Now we know that at infinite dilution 
the whole of the sodium chloride is dissociated. At finite dilutions, 
therefore, the degree of dissociation, i.e. the proportion of the whole 
which exists in the state of ions, is equal to the quotient of the 
molecular conductivity at the dilution considered by the molecular 
conductivity at infinite dilution. The degree of dissociation is generally 
denoted by m } so we have the equation 

m= — . 

For example, if we wish to ascertain the degree of dissociation of 
sodium chloride at a dilution of 1 litre, i.e. in normal solution, we 
divide the molecular conductivity, 69*5, by the molecular conductivity 
at infinite dilution, viz. 102*9, and obtain as quotient 0*675. In a 
normal solution of sodium chloride at 18°, a little over two-thirds of 
the salt is split up into its constituent ions. 

It must be very specially emphasised that the molecular con- 
ductivity itself is no measure of the degree of dissociation of a 
dissolved substance ; the true measure is the ratio of this conductivity 
to the molecular conductivity at infinite dilution. The degree of 
dissociation is not proportional to the molecular conductivity unless 
under certain conditions which must be carefully specified. For 
example, the molecular conductivity of a normal solution of sodium 
chloride at 50° is 120. This is nearly twice as great as the molecular 
conductivity at 18°, but the increase cannot come from a duplication 
of the number of ions, as the salt at 18° is already more than half 
dissociated. The great increase in the molecular conductivity is due 
to the increase in the other factor, namely the speed of the ions. The 
fluid friction of the solution is greatly diminished by the rise in 
temperature, and consequently the ions move much faster, thus in a 
given time conveying more electricity. At 50° the molecular 
conductivity at infinite dilution is 185. If we therefore divide 120 
by this number we obtain the degree of dissociation, approximately 
equal to 0*65, which is practically the same value as we got for 18°. 
In general, we find with salts that rise of temperature, while greatly 
augmenting the molecular conductivity, has very little effect on the 
degree of dissociation, the increase in the conductivity being almost 
wholly due to the increased speed of the ions. 

In a similar way the addition of non-conducting substances to salt 
solutions lowers the conductivity without appreciably altering the 


degree of dissociation as measured by the ratio of the molecular 
conductivity at the dilution considered to that at infinite dilution. 
Thus diethylammonium chloride dissolved in water and in mixtures 
of water and ethyl alcohol gave the following values at 25° for 
decinormal solutions, and for infinite dilution : — 

ntage of Alcohol 


by Volume. 



m= — ■ 

























The column headed ^ gives the effect of the alcohol in reducing the 
speed of the ions, since at infinite dilution the salt is entirely dissociated 
in each case, the number of ions being therefore the same throughout. 
In the last column we have the degree of dissociation as measured 
by the ratio of the molecular conductivity at 10 litres to //,«,. The 
addition of alcohol diminishes both the speed of the ions and the 
dissociation, so that the molecular conductivity in decinormal solution 
is reduced from both these causes. It will be noticed, however, that 
these two effects of the addition of alcohol do not go hand in hand. 
The degree of dissociation is scarcely affected by the first substitutions 
of alcohol for water (up to 30 per cent), while the speed of the ions, 
and consequently the molecular conductivity, is reduced to one-half. 
On the other hand, when nearly all the water has been replaced by 
alcohol, the effect of further additions is scarcely noticeable on the 
speed of the ions, but very marked on the degree of dissociation, and 
consequently on the molecular conductivity, in decinormal solution. On 
the whole the molecular conductivity in decinormal solution in 90 
per cent alcohol is only about a fifth of what it is in pure water : if 
the speed of the ions alone had been affected, the reduction would 
have been to a value a little more than a third of the value for water ; 
the balance of the reduction is due to diminution in the degree of 

The student is particularly recommended to a close study of the 
above examples, in order that he may become familiar with the two 
factors on which the molecular conductivity depends, as beginners 
almost invariably neglect to take account of the change in speed of 
the ions under different conditions, and thus from the values of the 
molecular conductivity draw utterly erroneous conclusions regarding 
the degree of dissociation. Degree of dissociation is never proportional 
to molecular conductivity unless the speed of the ions is the same in 
the two solutions compared. If two dilute solutions contain the same 
salt dissolved in the same solvent at the same temperature, then the 
degree of dissociation of the substance in the two solutions is propor- 
tional to the molecular conductivity, for the maximum molecular 



conductivity is the same in both cases. But if in the solutions compared 
the dissolved substance is different, the solvent is different, or the 
temperature is different, then the molecular conductivity is no longer 
a measure of the degree of dissociation, for the maximum molecular 
conductivity will no longer be the same. 

If we investigate the influence of dilution on the molecular con- 
ductivity of dilute solutions, we find that the weak acids and bases 
which form the group of half -electrolytes obey a law which was 
deduced by Ostwald from theoretical considerations, as will be shown 
in a subsequent chapter. Since other conditions are the same, and 
increase in dilution does not affect the speed of the ions, the change 
in the molecular conductivity observed is due entirely to change in 
the degree of dissociation. If we represent the degree of dissociation 
by m and the dilution by % the following relation holds good : — 

(1 - m)v 

= constant. 

The constant is usually denoted by h, and is called the dissociation 
constant. In the subjoined tables are given the values obtained at 
25° for acetic acid and ammonia respectively. 

Acetic Acid, CH 3 COOH 
Moo =364 












Mean . 

. 0-0018C 


NH 4 (OH) 

Moo = 

= 237 






























In the third column of the tables is given the percentage dis- 
sociation, i.e. the degree of dissociation multiplied by 100; and in the 
fourth column is one hundred times the value of the dissociation 
constant derived from the above formula. This centuple constant is 


often used instead of the smaller number on account of its leading for 
all substances to more convenient figures. The values of the constant 
at particular dilutions vary as a rule only 1 or 2 per cent from 
the mean, and this variation is due to errors of observation, which are 
greatly magnified in the calculation of the constant. 

In the first place, it is evident from the tables that acetic acid and 
ammonia in equivalent solutions are about equally dissociated, the 
former into hydrogen, H\ and acetions, CH 3 COO', the latter into 
ammonium, NH" 4 , and hydroxyl, OH'. If we compare these tables 
with that given on p. 221 for a good electrolyte, it will be seen that 
dilution has a far greater influence on the molecular conductivity in 
the former case than the latter. For sodium chloride an increase of 
the dilution from 1 to 100,000 only increases the molecular con- 
ductivity by about half its value ; while an increase in the dilution 
from 8 to 1024 increases the molecular conductivity of acetic acid 
more than tenfold. Although the molecular conductivity thus 
increases much more rapidly with the dilution than is the case for 
good electrolytes, yet the increase is not nearly proportional to the 
increase in the dilution. "When the degree of dissociation is small the 
molecular conductivity is roughly proportional to the square root of 
the dilution, as may be seen from a consideration of the general 

m 2 

71 — — r = k 
(1 -m)v 

For small degrees of dissociation, 1 -m is not greatly different from 1, 
so that the formula becomes 

m 2 = lev, or m = It J v. 

If in the general formula we put m = 0*5, i.e. if we assume that 
the electrolyte is half dissociated, we obtain a conception of the 
physical dimensions of the constant k. The expression becomes 

0-5 2 

(1 - 0-5> ' 

whence — = Jc. or -c = h if c = — • 

2v 2 v 

In words, the dissociation constant h is numerically equal to half the 
concentration at which the substance is half dissociated, the concen- 
tration being expressed in gram molecules per litre. Thus for acetic 
acid we have £ = 0*000018, whence c = 0*000036, i.e. acetic acid is 
half dissociated when the concentration of its aqueous solution is 
0'000036th normal. 

It is obvious that if in such a solution, which contains only about 
2 parts of acetic acid per million, the acid is only half dissociated, the 




direct determination of the molecular conductivity for solutions in 
which the acid is wholly dissociated is an impossibility. Yet the 
value of the molecular conductivity must be known in order that the 
degree of dissociation may be calculated. It has therefore to be 
determined indirectly by means of Kohlrausch's Law. Although weak 
acids and weak bases are but half-electrolytes, their salts are good 
electrolytes, and as much dissociated in solution as the corresponding 
salts of strong acids and bases. Thus for sodium acetate at 25° we 
have the following numbers : — 
























For ammonium chloride, Kohlrausch found at 18° numbers much the 
same as those for sodium chloride, p. 221, viz. — 
































It is an easy matter, then, to find numbers for the molecular con- 
ductivity at infinite dilution in the case of salts of weak acids or 
bases. Now, according to Kohlrausch's Law, there is a constant 
difference between the maximum molecular conductivities of all acids 
and their sodium salts, a difference due to the difference in speed of 
the hydrogen and sodium ions. But strong acids like hydrochloric 
acid are at equivalent dilutions quite as much dissociated as their 
sodium salts, so that their maximum molecular conductivities may be 
determined experimentally. For these acids we can thus get directly 
the difference between their maximum molecular conductivity and 
that of their sodium salts. This difference, amounting at 25° to about 
275 in the customary units, when added to the maximum molecular con- 
ductivity of the sodium salt of an acid, which can always be directly 
determined, gives the molecular conductivity of the acid itself at 
infinite dilution, and this enables us to give the degree of dissocia- 
tion of the acid at any other dilution. 

It is a curious fact, of which no adequate explanation has as yet been 
given, that good electrolytes do not obey Ostwald's dilution law, 


which holds so accurately for the half-electrolytes. Certain empirical 
relations have, however, been found connecting the degree of 
dissociation and the dilution, and these have a form similar to that of 
Ostwald's dilution formula, although they have not the theoretical 
foundation possessed by the latter. The first of these relations is 
called Rudolphi's dilution formula, and differs from Ostwald's by 
the square root of the dilution being introduced instead of the dilution 
itself. It has thus the form 

(1 -m) sj v 

- = constant. 

and agrees fairly well with the observed values. Thus for ammonium 
chloride at 18° we have the following numbers: — 
































Mean . .1*51 

The second empirical dilution formula is that of van 't Hoff, which 
in some respects is simpler than Kudolphi's, and accords quite as 
closely with the facts. We may write Ostwald's dilution formula in 
the form 

(-) 2 

- — — = constant, 
1 - m 

where — is the concentration of the dissociated portion of the 

2 fffi 

electrolyte, and the concentration of the undissociated part. If 

C d and C u represent these concentrations, we have the simple 

C£ , , 

-^- = constant. 

Rudolphi's formula gives no such simple relation of the concentrations 
of the dissociated and undissociated portions. Yan 't Hoff proposes 
the expression 

(1 - m) \/v 

- = constant, 

which may be written in the form 



; — — = constant. 

Here again we have a simple relation between the concentration of 
the dissociated and undissociated portions, and the constancy of the 
expression is at least equal to that obtained with Kudolphi's formula, 
as is shown by the subjoined table for ammonium chloride : — 



(1 - m)\/v 

























5000 0*991 1-54 

Mean . . 1*56 

According to the foregoing hypothesis of electrolytic dissociation, 
aqueous solutions of salts, strong acids, and strong bases have a 
certain proportion of the dissolved molecules split up into charged 
ions, and the amount in any given case is determinable from 
measurements of electrical conductivity. The ions are supposed 
to be independent of each other, and ought to act as separate 
molecules if the independence is complete. We should therefore 
expect that salt solutions, when investigated by the customary method 
of molecular-weight determination, should exhibit exceptional behaviour, 
and give for the molecular weights of the dissolved substances, values 
smaller than those which we should deduce from the ordinary 
molecular formulae. As has already been indicated, such abnormal 
values are frequently observed. When the molecular weight of 
sodium chloride and other similar salts is determined from the 
freezing or boiling points of their aqueous solutions, the numbers 
obtained are only equal to little more than half the values given 
by the formula NaCl, i.e. the depressions of the freezing point 
and the elevations of the boiling point have almost twice the 
normal value. A normal solution of cane sugar freezes at 
-1*87°; a normal solution of sodium chloride freezes at -3*46°. 
Such an abnormally high value for the depression indicates that there 
are more dissolved molecules in the normal solution of common salt 
than there are in the normal solution of sugar, although each solution 
in the usual system of calculation is supposed to contain 1 gram 


molecule per litre. It should be noted at once that it is only 
electrolytic solutions which show these abnormally high values, the 
values given by non - electrolytic solutions being almost invariably 
equal to, or less than, the normal values. Abnormally small values 
we have already attributed to molecular association (cp. p. 193), so 
that we ought, by parity of reasoning, to attribute the abnormally 
large values for salt solutions to a dissociation of the molecules. 

Van 7 t Hoff introduced for salt solutions a coefficient i which repre- 
sents the number by which the normal value of the freezing point, 
etc., must be multiplied in order to give the value actually found. 
Thus for the solution of sodium chloride we have the depression 3*46°, 
instead of the "normal" value 1*87 shown by solutions of non- 
electrolytes containing 1 gram molecule per litre. In this case 
i = 3*46/1*87 = 1*85, i.e. the normal value 1*87 has to be multiplied 
by 1*85 in order to bring it up to the observed value 3*46. 

In the first place, it is to be noticed that for salts, acids, and bases 
which, according to the theory of electrolytic dissociation, split up into 
two ions, the coefficient i is never greater than 2. If dissociation into 
two ions were complete, the value for the depression of freezing point, 
etc., would be twice the normal value, since each molecule represented 
by the ordinary chemical formula becomes two independent molecules 
by dissociation. Now at common dilutions the dissociation is never 
complete, so that the value of the depression should be something less 
than 2, as is actually found by observation. When the dissociation 
hypothesis admits of a dissociation into more than two charged 
molecules, the depressions of the freezing point give values of i 
greater than 2. Thus strontium chloride, according to the dissoci- 
ation hypothesis, splits up at infinite dilution into the three ions, 
Sr", CI 7 , and Cr, the first of which has two charges of positive electricity, 
and the others a charge each of negative electricity. At moderate 
dilutions, therefore, we ought to expect a value of i less than 3, but 
probably greater than 2, as the dissociation of salts is generally high. 
We find in accordance with this that strontium chloride in decinormal 
solution gives the depression 0*489, instead of the value 0*187 obtained 
for non-electrolytes. The value of i is therefore 0*489-*- 0*187 = 2*6. 

It is obvious from the above examples that there is a direct 
numerical relation between the degree of dissociation in any given case 
and van 't Hoff' s coefficient i 9 so that if one is given the other can be 
calculated. If the degree of dissociation of a salt in solution is m, 
the dissociation being into two ions, then there will be present in the 
solution 1 - m undissociated molecules and 2m dissociated molecules 
in all 1 + m molecules for each molecule represented by the chemical 
formula. The depression of the freezing point, etc., will therefore 
have 1 + m times the normal value, i.e. 

i = 1 + m. 


Should the original molecule split up into n ions when the dissociation 
is complete, m again representing the dissociated proportion, there will 
be present 1 - m undissociated molecules and nm dissociated molecules, 
in all 1 + (n - l)m molecules for each original molecule, i.e. we shall 

i- 1 + (n- l)m. 

Giving m in terms of % we have 

m-i- 1 

for dissociation into two ions, and 


ro = - 


for dissociation into n ions. 

A comparison of the values of i deduced for the same solution 
directly from the freezing point, and indirectly from determinations 
of m by means of the electric conductivity, shows very fair accordance 
between the two methods. In the first place, it is found that in the 
case of non-electrolytes where m= 0, we have i — 1, i.e. we obtain the 
normal value for the freezing-point depression, etc. ; and Arrhenius 
showed from his own freezing-point determinations that the values 
obtained for i with electrolytes differ from the corresponding values 
calculated from the electric conductivity by not more than 5 per cent 
on the average. Although this difference is comparatively great, 
better accordance was scarcely to be expected owing to the difficulty 
in effecting the comparison. It must be borne in mind that the 
molecular conductivity only gives accurate figures for the degree of 
dissociation when the solutions are so dilute that further dilution does 
not sensibly change the speed of the ions. Now this condition necessi- 
tates a dilution of at least 10 litres, i.e. the solution must not be more 
concentrated than decinormal, even in the most favourable case. But 
the normal depression for a decinormal solution is only 0*187°, so 
that to obtain an accuracy of 1 per cent on the value of i, as 
measured by the freezing-point depression, we must be able to 
determine depressions with an accuracy of about a thousandth of a 
degree centigrade. It is by no means easy to attain this degree of 
accuracy, the error in ordinary careful work with Beckmann's apparatus 
being nearer a hundredth of a degree than a thousandth. For more 
dilute solutions the relative error on the depression is of course still 
greater, and extraordinary precautions have to be taken if the freezing 
points are to be of any value in calculating van 't HofFs coefficient i, 
or the degree of dissociation m. The following table gives a comparison 
of the degree of dissociation of solutions of potassium chloride as 
calculated from Kohlrausch's determinations of the conductivity, and 
from the mean value of the best series of observations on the freezing 


point by different investigators. In the first column we have the 
dilution in litres, in the second the percentage dissociation deduced 
from the freezing points, and in the third the same magnitude calculated 
I by means of the dissociation hypothesis from Kohlrausch's observations 
of the conductivity : — 

Dilution. Percentage Dissociation — 100m. 

v From Freezing Point. From Conductivity. 

10 86-0 86-2 

20 88-8 89*1 

33-3 90*8 91*1 

100 95*3 94*4 

This is a rather favourable example of the close numerical agreement 
attained by the two methods of calculation, which, it must be re- 
membered, are entirely independent of each other. As a rule the 
divergencies amount to as much as 2 per cent of the actual values, 
but the parallelism between the series is always very close. 

From what has been stated above, it is obvious that the hypothesis 
of electrolytic dissociation affords a satisfactory explanation of the 
anomalous behaviour of the solutions of salts, strong acids, and strong 
bases with regard to freezing point, boiling point, and in general all 
magnitudes directly derivable from the osmotic pressure. Not only 
does it do this, but it explains also very directly the additive character 
of most of the properties of salt solutions. In a previous chapter it 
has been shown that the properties of salts in aqueous solution are 
most simply explained when we assume that they are additively 
composed of two terms, one depending on the basic or positive portion 
of the salt, and the other on the acidic or negative portion. Accord- 
ing to the theory of electrolytic dissociation, the salt is actually 
decomposed on dissolution in water into its positive and negative 
ions, which then lead an independent existence, each conferring 
therefore on the solution its own properties undisturbed by the 
properties of the other ion. The total numerical value of any given 
property in a salt solution will therefore be made up of the value 
for the positive ion plus the value for the negative ion, at least when 
the solutions considered are dilute, and so the additive character of the 
properties of salt solutions is satisfactorily accounted for. 

The theory of electrolytic dissociation offers, too, a simple 
explanation of a set of facts that are so familiar that we generally 
accept them without any attempt at accounting for them. It is well 
known that salts in aqueous solution enter into double decomposition 
with the greatest readiness. If we add a soluble silver salt to a soluble 
chloride, a precipitate of silver chloride is at once produced. The 
positive and negative radicals here exchange partners, and they do so 
readily because the positive and negative for the most part exist free 
in the solution as ions, so that the whole action practically consists of 
the union of the silver ions with the chloride ions to produce the 


insoluble silver chloride. If alcohol is used as solvent, the action takes 
place with equal readiness. If we take now a solution in alcohol of 
an organic chloride such as phenyl chloride, we find that it is a non- 
electrolyte, and corresponding with this, it may be mixed with an 
alcoholic solution of silver nitrate without any double decomposition 
taking place. Even after boiling for a considerable time there is 
little or no silver chloride produced. In this case there are no free 
chloride ions in the solution to unite directly with the silver ions, 
and consequently the action is much slower. Other organic halogen 
compounds act much more rapidly than phenyl chloride when in 
alcoholic solution, but it is very doubtful if any act with a speed 
even approximately comparable with that of the inorganic chlorides. 
Reactions between salts in aqueous solution, which do not proceed 
by a simple rearrangement of the ions, are much slower than 
the double decompositions where the ions undergo no change except 
rearrangement. We have instances of this type of reaction in the 
oxidations and reductions occurring in aqueous solution, such as the 
conversion of ferrous into ferric salts, or stannous into stannic salts, 
and vice versa. Although one may find exceptions to these rules, 
they are yet of a general character such as we should expect on the 
theory of electrolytic dissociation. 

It has occasionally been urged that the existence of chlorine in a 
solution of sodium chloride cannot be accepted even hypothetically, 
as the solution shows none of the properties of a solution of chlorine. 
This, of course, rests on a misunderstanding. What we suppose to 
exist in the solution is not chlorine, but the chloride ion. The 
molecule of the former consists of two uncharged atoms of chlorine, 
the molecule of the latter consists of one atom of chlorine charged 
with positive electricity according to Faraday's Law, i.e. every 35 '5 g. 
of chlorine as ion has 96,500 coulombs of electricity associated with it. 
But we know that a charge of electricity profoundly affects the chemical 
properties of substances. The mere fact of the chemical changes 
accompanying electrolysis is evidence of this, and other instances 
exist in plenty. Neither aluminium nor mercury decomposes water 
at the ordinary temperature, but if the aluminium is coated with 
mercury the amalgam formed has this power, hydrogen being evolved 
and aluminium hydroxide produced. The action of the copper-zinc 
couple is similar. These metals separately are unable to perform 
chemical reactions which are easily brought about by zinc coated with 
copper. Another familiar instance is the behaviour of zinc towards 
sulphuric acid. Commercial zinc readily decomposes sulphuric acid 
with evolution of hydrogen, whilst pure zinc is almost without action 
on the dilute acid. This is due to the fact that the commercial metal 
contains other metals as impurities, and these increase the action of 
the zinc. If to the solution of sulphuric acid in contact with pure 
zinc we add a small quantity of a platinum or a cobalt salt, these 


metals are deposited on the surface of the zinc and vigorous action 
ensues. In each of these cases the two metals on coming into contact 
assume electrical charges which greatly modify their ordinary chemical 

The student who wishes to pursue the subject of Electrolysis and Electro- 
chemistry from the dissociation point of view may be referred to M. Le 
Blanc, Electrochemistry ; and R. Lupke, Electrochemistry. 



Many of the chemical actions familiar to the student in his laboratory 
experience are reversible, i.e, under one set of conditions they proceed 
in one direction, under another set of conditions they proceed in the 
opposite direction. Thus if we pass a current of hydrogen sulphide 
into a solution of cadmium chloride, double decomposition occurs, 
according to the equation 

CdCl 2 +H 2 S = CdS + 2HCl. 

If the precipitate is now filtered off and treated with a solution of 
hydrochloric acid of the requisite strength, the action proceeds in the 
reverse direction, the equation being 

CdS + 2HCl = CdCl 2 + H 2 S. 

What determines the direction of the action in this case is apparently 
the relative quantities of hydrogen sulphide and hydrogen chloride 
present in the solution. If hydrogen sulphide solution is added to a 
solution of a cadmium salt which contains a considerable quantity of 
free hydrochloric acid, a part only of the cadmium will be precipitated 
as sulphide, part remaining as soluble cadmium salt. We are here 
dealing with a balanced action, and we shall find it convenient to 
formulate actions of this type by means of the ordinary chemical 
equation for the action with oppositely-directed arrows instead of the 
sign of equality, thus : — 

CdCl 2 + H 2 S^CdCl 2 + H 2 S. 

As a rule, in analytical work we do not wish to stop half-way in a 
chemical action, and therefore choose such conditions for the action 
that it will proceed practically to an end. Cadmium chloride is 
completely precipitated as sulphide when there is little hydrochloric 
acid present, and when a considerable excess of sulphuretted hydrogen 
has been added. 

chap, xxn BALANCED ACTIONS 235 

A balanced action of the same sort, but with the point of balance 
or equilibrium towards the other end of the reaction, is to be found 
when a solution of hydrogen sulphide is added to a solution of nickel 
or cobalt chloride. A black precipitate of the metallic sulphide is 
formed, but even though a very large quantity of hydrogen sulphide 
is present, the precipitation is never complete. A very moderate 
quantity of free hydrochloric acid will prevent the precipitation 
altogether. It is thus possible, by suitably choosing the conditions, 
to effect a separation of cadmium from nickel or cobalt by means of 
sulphuretted hydrogen, although in the two cases we are dealing with 
balanced actions of precisely the same type. If some dilute hydro- 
chloric acid is added to the solution of the mixed metallic chlorides, 
hydrogen sulphide will precipitate the cadmium almost completely, 
and precipitate practically none of the nickel or cobalt salt, even 
though it is present in great excess. 

If ammonium hydroxide solution is added to a solution of a 
magnesium salt, say magnesium chloride, part of the metal is 
precipitated as magnesium hydroxide, in accordance with the equa- 

MgCl 2 + 2NH 4 OH = Mg(OH) 2 + 2NH 4 C1, 

but the precipitation is never complete, for the reverse action 
Mg(OH) 2 + 2NH 4 C1 = MgCl 2 + 2NH 4 OH 

occurs simultaneously, and a state of balance results. In analytical 
practice we have usually excess of ammonium chloride present from 
the beginning, so that the addition of ammonium hydroxide produces 
no precipitate in the solution of magnesium salt. The reverse action 
can be easily studied by shaking up freshly-precipitated magnesium 
hydroxide with a solution of ammonium chloride, when the magnesium 
hydroxide will be found to dissolve. 

We may consider such cases of chemical equilibrium from the 
same standpoint as we adopted in Chapter X. for physical equilibrium. 
In the last instance given above, viz. the addition of ammonium 
hydroxide solution to magnesium chloride solution, if we mix definite 
amounts of solutions of definite strengths, the precipitation of 
magnesium hydroxide will come to an end when the reaction has 
proceeded to a certain ascertainable extent. Equilibrium is then 
reached, and the system undergoes no further apparent change. We 
may still conceive the opposed reactions to go on as before, but at 
such a rate that exactly as much magnesium hydroxide is formed by 
the direct action as is reconverted into magnesium chloride by the 
reverse action. Now at the beginning of the action there is no 
magnesium hydroxide or ammonium chloride present at all ; these are 
only formed as the direct action proceeds. There is therefore at the 
beginning no reverse action. It is obvious that if a state of balance 


is to be reached, the rate at which the direct action proceeds must fall 
off, or the rate at which the reverse action proceeds must increase, or 
finally both these changes may occur together. As the action 
progresses, the relative proportions of the reacting substances and the 
products of the reactions vary, and so we should be disposed to connect 
the actual rate at which magnesium hydroxide is formed or decom- 
posed with the amounts of the different substances in solution. 
Further information on this point may be got by adding ammonium 
chloride to the solution of magnesium chloride from the beginning. 
If we add a very small quantity of ammonium chloride before we 
add the ammonium hydroxide, keeping the relative proportions of 
the other substances the same as before, we shall find that the 
amount of magnesium hydroxide precipitated is smaller than before, 
and if we go on increasing the amount of ammonium chloride little by 
little, we at last reach a point when no magnesium hydroxide is 
precipitated at all. Evidently, then, the presence of ammonium 
chloride favours the reverse action, and that in a manner proportionate 
to the amount added. Increase in the amount of ammonium 
hydroxide, on the other hand, favours the direct action, so that on the 
whole we should be inclined to suspect that the rate of a reaction 
depended on the amount of reacting materials present. 

Guldberg and Waage, from a consideration of many experiments, 
formulated the connection between rate of action and amount of 
reacting substances in the following simple way. The rate of chemical 
action is proportional to the active mass of each of the reacting 
substances. This rule must in the first instance be taken to apply to 
solutions or gases, for it is in their case only that " active mass" can be 
properly defined. By active mass Guldberg and Waage understood what 
we usually term the molecular concentration of a dissolved or gaseous 
substance, i.e. the number of molecules in a given volume, or in the 
ordinary chemical units, the number of gram molecules per litre. It 
is possible, however, as Arrhenius has suggested, that this molecular 
concentration in solution is not really a measure of the active mass, and 
that instead of it we ought to substitute the osmotic pressure of the 
dissolved substance. For our present purpose, we may take the active 
mass to be proportional to the molecular concentration without 
risk of committing any serious error, for, as we have already seen, 
there is, at least in dilute solution, almost exact proportionality 
between osmotic pressure and molecular concentration. 

Suppose we are dealing with the following chemical action — 

A+B = C + D, 

where the letters represent single molecules of the substances as in 
ordinary chemical formulae. Let the molecular concentrations of the 
original substances A and B be a and b respectively. The rate of the 
reaction will then, according to Guldberg and Waage, be proportional 


to a and also proportional to b, i.e. it will be proportional to the product 
ab. At the beginning of the action, then, we have the expression 

Eate = db x constant, 

if by rate of reaction we mean the number of gram molecules of each 
of the reacting substances transformed in the unit of time, usually the 
minute. The constant, therefore, in the above equation, which is 
generally denoted by k t represents the rate at which the action would 
proceed if each of the reacting substances were at the beginning of the 
reaction of the molecular concentration 1, as may easily be seen from 
the transformed equation 


It must be remembered that as the action progresses the concentration 
of both A and B will fall off, and that the rate at which these 
substances are transformed into the substances G and D must there- 
fore progressively diminish. If at the time t the concentration of A 
has diminished by x gram molecules per litre, the concentration of B 
will have diminished by a similar amount, and the rate at which these 
substances are then transformed will be 

Rate^ = k(a - x)(b - x). 

The constant k is still the same as in the preceding equation, and has 
still the same significance, as may be seen from a consideration of 
the formula. It is indeed characteristic of the reaction, being inde- 
pendent of the concentrations of the reacting substances, although 
variable with temperature, nature of the solvent, etc. It is customary 
to call it the velocity constant of the action, and the student is 
recommended to bear in mind that it is the rate at which the 
reaction would proceed were the reacting substances originally present, 
and constantly maintained, at unit concentration. 

If the reaction A + B=C + D is not reversible, the transformation 
of A and B into G and D will go on at a gradually decreasing rate 
until at least one of the reacting substances has entirely disappeared. 
If the action, on the other hand, is reversible, the transformation of 
G and D into A and B will begin as soon as any of the former 
substances are formed by the direct action. If we suppose no G and 
D to be present when the action commences, we have at the beginning 
of the direct action c = and d = 0. At the time t, when x of the 
original products has been transformed, we have c = x and d = x. If 
k' is the velocity constant of the reverse action, then at the time t 

Rate^ = k'x 2 . 

Now after the direct action has proceeded for a certain time, which we 


may call t, a state of equilibrium sets in. Let the diminution of the 
molecular concentration of A and B have at that time the value £, 
then the rate of the direct transformation is 

The rate of the reverse transformation at the same time is 

k'£ 2 . 

But these rates must be equal if the system is to remain in equilibrium, 
for in a given time as much of A and B must be formed by the 
reverse action as disappear in the same time by the direct action. 
The equation 

consequently holds good, and this when transformed gives 
\ *£ ^ = = constant. 

Since k and k' are independent of the concentration, their ratio is also 
independent of the concentration, so that we have for equilibrium the 
product of the active masses of the substances on one side of the 
chemical equation, when divided by the product of the active masses 
of the substances on the other side of the chemical equation, giving a 
constant value no matter what the original concentrations may have 
been. If we wish to make the formula perfectly general, we may 
suppose that the concentrations of the products of the direct action 
were c and d respectively. At the point of equilibrium the concen- 
trations of these substances will then be c + £ and d + £, so that the 
constant magnitude will be 

( «-£)( *-£)# 

When, as very frequently happens, the original substances are taken 
in equivalent proportions, and none of the products of the direct 
action are present at the beginning, the constant quantity has the 
simple form 

? ' *' 

in which a represents the original molecular concentration of both 
reacting substances. 

No good practical instance of the application of the above formula 
is known, although approximations to it are in some cases obtainable. 
The best is perhaps the equilibrium between an ethereal salt, water, 
and the acid and alcohol from which the ethereal salt is produced. 
Thus if we allow acetic acid and ethyl alcohol to remain in contact, 
they will interact with production of ethyl acetate and water. The 


action, however, will not be complete, because the reverse action will 
simultaneously take place, the ethyl acetate being decomposed by the 
water into acetic acid and ethyl alcohol. The equation for the reversible 
action is 

CH 3 . COOH + C 2 H 5 . OH X CH 3 . COOC 2 H 5 + H 2 0. 

If the acetic acid and ethyl alcohol are taken in equivalent proportions, 
the action ceases when two-thirds of the substances have been trans- 
formed into ethyl acetate and water. Supposing the active mass of 
the acid and alcohol to have been originally 1, the active masses at 
equilibrium will be 

Acetic acid = 1 - § = J, 
Alcohol = 1 - § = |, 

Ethyl acetate = f , 
Water = f , 

and the constant quantity will be 

(i-g)' _W_i 

This constant holds good for any proportions of the reacting substances, 
being the ratio of the velocity constants of the opposed reactions. 
We can therefore use it to determine the extent to which a mixture 
in any proportions of acetic acid and ethyl alcohol will be transformed. 
Thus, if for 1 equivalent of acid we take 3 equivalents of alcohol, 
at what point will there be equilibrium ? If £ represents the amount 
transformed at equilibrium, we have 

whence £ = 0*9, i.e. 90 per cent of the acid originally taken will be 
converted into ethereal salt by 3 equivalents of alcohol against 66*6 
converted by 1 equivalent of alcohol. Direct experiment has shown 
that this amount of the acid is actually transformed. In this instance 
the substances are merely in solution in each other, but the presence 
of a neutral solvent, such as ether, in no way affects the equilibrium, 
although it greatly reduces the speed of the opposed reactions. 

There is an example of equilibrium in aqueous solution which has 
been verified with the utmost strictness for a very large number of 
substances, namely, the equilibrium between the ions of a weak acid 
or base and the undissociated substance itself. Suppose the substance 
considered to be acetic acid. If a gram molecule of the acid is dissolved 
in water so as to give v litres of solution, its active mass is l/v in our 
units. No sooner is the acid dissolved than it begins to split up 
into hydrogen ions and acetic ions. The rate at which this action 


progresses is, according to the principle of mass action, proportional 
to the active mass of undissociated acetic acid present at the time 
considered. Let the proportion of acetic acid dissociated be m. The 
proportion undissociated will therefore be 1 - wi, with the active mass 
I — tfli 

— . The rate of dissociation will thus be 

1 -m 

c 1 


where c is the velocity constant of the dissociation. But the action 
is a balanced one, so that undissociated acetic acid will be reformed 
from the ions at the same time as the dissociation proceeds by the 
direct action. When the proportion of the acid transformed into 
ions is m, each of the ions will have the active mass m/v, and thus 
we shall have for the rate of reunion 


if c is the velocity constant of this reaction. Suppose now, in 
particular, that m is the proportion decomposed into ions when 
equilibrium has taken place. The rates of the opposed reactions 
must then be equal, and we thus obtain the equation 

1 -m 

= c 


m _ ° 

' — ~ = rC. 

(1 - m)v c 

This is Ostwald's dilution formula referred to on p. 224, and it was 
first deduced by him from the mass-action principle of Guldberg and 
Waage. The experimental verification has been given for acetic acid 
and ammonia (p. 224). What holds good for them is equally true 
for many hundreds of feeble acids and bases which have been in- 

It has already been indicated that this dilution law does not hold 
good for salts or for powerful acids and bases, and we must leave it 
meantime an open question whether the principle of mass action, the 
active mass being measured by the molecular concentration, applies to 
them or not. It would appear not to do so, and we are as yet without 
data to explain the divergence. Dilution laws similar to those of van 't 
Hoff and Rudolphi can be obtained by assuming that the active mass 
is measured by a power of the molecular concentration other than the 
first, fractional powers being admissible. Such assumptions are, of 
course, purely empirical, and are not applicable to all known cases. 
Their value is therefore at present rather on the practical than the 
theoretical side. 



The principle of mass action, which has been found above to hold 
good for substances in solution, also holds good for substances in the 
gaseous state. Nitrogen peroxide, when dissolved in chloroform, 
dissociates into simpler molecules (p. 194), according to the equation 

N 2 4 t- N0 2 + N0 2- 

This is quite analogous to the equation for the dissociation of a weak 
acid or base, the only difference being that the products of dissociation 
are here of the same kind instead of being of different kinds. If m 
represents the dissociated proportion, 1 - m the undissociated propor- 
tion, and v the dilution, we have, as before, the following expression 
regulating the equilibrium — 

m 2 

: constant, 

(1 - m)v 

and this has been proved to be in accordance with the observed facts. 
Now nitrogen peroxide is also known in the gaseous state, and 
dissociates under these conditions precisely in the same manner as 
when it is dissolved in such a solvent as chloroform. The chemical 
equation for the dissociation is the same as before, and so is the 
expression for the amount of dissociation. The dilution v, in this 
case, is the volume occupied by 1 gram molecule of the gaseous 
substance. In cases of dissociation proper, i.e. cases of balanced action 
into which gaseous substances enter, the amount of dissociation is most 
readily determined by ascertaining the pressure and density. With 
a given amount of substance occupying a certain volume at a known 
temperature it is easy to calculate what pressure it will exert, accord- 
ing to Avogadro's principle, if there is no dissociation of the molecules. 
If there ,is dissociation, the pressure will, coeteris paribus, be greater, for 
in the given space there are more molecules than would be if no 
dissociation had occurred. With nitrogen peroxide, simultaneous 
determinations of the pressure and density are made at constant 
temperature, both being varied by changing the volume, and from 
these the magnitudes entering into the above formula can be calculated. 
In this case also there is close agreement between the observed 
numbers and the numbers calculated from the formula. 

Considering the matter from the kinetic molecular point of view, 
it is obvious that the position of equilibrium will in some cases depend 
on the volume occupied by the system. Dilution increases the degree 
of dissociation of electrolytes in aqueous solution; and dilution 
increases the dissociation of nitrogen peroxide, whether in the state of 
solution or in the gaseous state. Each molecule of undissociated 
material decomposes on its own account, and is independent of the 
presence of other molecules. The number of undissociated molecules, 
therefore, which will decompose in a given time is entirely unaffected 



by the volume which they may be made to occupy. Dilution then 
does not affect the number of molecules which will dissociate in a 
given time. But dilution does affect the number of molecules reformed 
from the dissociated products in a given time, for here each dissociated 
molecule must meet with another dissociated molecule if an undis- 
sociated molecule is to be reproduced. Now the chance of two 
molecules meeting depends on the closeness with which the molecules 
are packed. If the particles are close together, they will encounter 
each other frequently ; if they are far apart, they will meet only rarely. 
Suppose that we double the volume in which a certain amount of 
nitrogen peroxide is contained. Each N0 2 molecule has on the 
average to travel twice as far as before in order to meet another 
N0 2 molecule. The number of encounters in a given time will there- 
fore be reduced to a fourth. The rate therefore of the reverse reaction 
corresponding to the equation 

N 2 4 £ 2N0 2 

is reduced to a fourth by doubling the dilution ; the rate of the direct 
action is unaffected. If therefore there was equilibrium between the 
direct and reverse actions before the system was made to occupy a 
larger volume, this equilibrium will be disturbed and a new equilibrium 
will be established at a point of greater dissociation. Here there will 
be proportionately less of the undissociated substance, and proportion- 
ately more of the products of dissociation, in order that the rates at 
which the undissociated nitrogen peroxide is decomposed and reformed 
may again be the same. 

In general, we may say that when dissociation is accompanied by 
an increase of volume at constant pressure, as is almost invariably 
the case, the extent of the dissociation is increased if we increase 
the volume in which the dissociating substance is contained. 

Sometimes the dissociation is not accompanied by change of volume 
of pressure, and then we find that neither pressure nor volume has 
any influence on the degree of dissociation. Strictly speaking, perhaps 
the term " dissociation " ought not to be applied to such cases at all, 
but in ordinary chemical language we almost always allude, for 
example, to the decomposition of hydriodic acid as a dissociation. 
The decomposition takes place according to the equation 

2HI t H 2 + I 2 , 

two volumes of hydriodic acid giving two volumes of decomposition 
products. Here, before hydrogen and iodine can be produced, two 
molecules of hydrogen iodide must meet, and before the hydrogen 
iodide can be reformed, a molecule of hydrogen must encounter a 
molecule of iodine. The chances of each kind of encounter will be 
equally affected by a change in the concentration, so that the equili- 
brium established for one concentration will hold good at any other 


concentration. Although, therefore, the velocities of the opposed 
reactions are altered by alteration in the concentration, they are 
altered to the same extent, and the position of equilibrium is unaffected. 
Since with gases we usually effect changes in concentration by changing 
the pressure, we should expect that change of pressure would have no 
influence on the dissociation equilibrium of hydrogen iodide. As a 
matter of fact, it has been found that increase of pressure rather 
increases the amount of dissociation, especially under certain conditions, 
but this may be due to the action not taking place strictly according 
to the above equation, or to the volumes of the substances on the two 
sides of the equation not being exactly equal, owing to a slight divergence 
from Avogadro's Law. 

The above instances of balanced action are all of such a type that 
the system in which the equilibrium occurs is a homogeneous system — 
either a homogeneous mixture of gases, or of substances in solution. 
We have now to consider cases of heterogeneous equilibrium, the 
system consisting of more than one phase. As an example, we may 
take the dissociation of calcium carbonate by heat, according to the 

CaC0 3 ;> CaO + C0 2 . > 

Here we are dealing with two solids and one gas. The active mass of 
the gas may either be measured by its concentration, i.e. density, or 
its pressure, the two being closely proportional. There is evidently a 
difficulty in expressing the active mass of a solid in a similar way. 
The pressure of a solid cannot be measured as the pressure of a gas 
can, and the active mass could scarcely be expected to be proportional 
to the density of the solid, which is the only direct meaning we can 
give to concentration in this connection. The most instructive way of 
looking at such an equilibrium is to imagine it to take place entirely 
in the gaseous phase, the solids simply affording continuous supplies 
of their own vapour. Every liquid has, as we have seen, a definite 
vapour pressure for each temperature. At 360° the vapour pressure 
of mercury is 760 mm. At the ordinary temperature it is not directly 
measurable, being too small, but the presence of mercury vapour over 
liquid mercury may easily be rendered evident at temperatures much 
below the freezing point. Ice, too, has a vapour pressure of a few 
millimetres at the freezing point, and there is no very good reason to 
think that this vapour pressure would disappear entirely at any 
temperature, however low, although it might become vanishingly 
small. We may then freely admit the possibility of calcium carbonate 
or calcium oxide vapour over the respective substances, although the 
pressure of these vapours is so small that they escape our means of 
measurement, or even of detection. If we are to assume the existence 
of these minute quantities of vapour, we must assume that the same 
laws are followed in their case as in those instances which are acces- 


sible to our measurement. In particular, we must assume that at a 
given temperature calcium carbonate, for example, will have a perfectly 
definite vapour pressure, which will remain constant as long as the 
temperature remains constant. We have thus in the gaseous phase a 
constant concentration or pressure of the vapour of the solid, the 
presence of the solid maintaining the pressure at its proper value, 
although the substance may be continually removed from the gaseous 
phase by the chemical action. This constant concentration is obviously 
unaffected by the quantity of solid present, as the vapour pressure of 
a small quantity is as great as that of a large quantity of the same 
solid. From the above reasoning, then, we conclude that the active 
mass of a solid is a constant quantity, being in fact proportional to the 
vapour pressure of the solid, which is constant for any given tempera- 
ture. Experiment confirms this conclusion, which was, indeed, arrived 
at by Guldberg and Waage on purely experimental grounds. 
For the action 

CaC0 3 ;> CaO + C0 2 , 

if P, F and p denote the equilibrium pressures (or active masses) of 
calcium carbonate, calcium oxide, and carbon dioxide respectively, we 
have at the point of equilibrium the following equation — 

in which all the quantities are constant except p. At any given 
temperature, then, the pressure of carbon dioxide over any mixture of 
calcium carbonate and calcium oxide has a fixed value quite inde- 
pendent of the proportions in which the solids are present. This 
particular pressure of carbon dioxide, which is called the dissociation 
pressure of the calcium carbonate, is the only pressure of carbon 
dioxide which can be in equilibrium with calcium carbonate, calcium 
oxide, and with any mixture of the two. Greater pressures are in 
equilibrium with calcium carbonate only, smaller pressures are in 
equilibrium with calcium oxide only. As the temperature increases, 
the dissociation pressure likewise increases, so that we can get a 
temperature curve of dissociation pressures resembling that for the 
vapour pressures of a liquid. 

Similar phenomena to the above are met with in the dehydration 
of salts containing water of crystallisation. The hydrated and the 
anhydrous salts are here the solid substances, and water vapour the gas. 
For a given temperature each hydrate has a definite dissociation 
pressure of water vapour over it. There is here, however, the 
complication that a salt usually forms more than one hydrate, in 
which case the dehydration often proceeds by stages, the hydrate 
with most water of crystallisation not passing immediately into water 
vapour and the anhydrous salt, but into water vapour and a lower 


hydrate. Take, for example, the common form of copper sulphate, 
the pentahydrate CuS0 4 , 5H 2 0. At 50° this hydrate gradually loses 
water (cp. Fig. 20, p. Ill), and becomes converted into the trihydrate 
CuS0 4 , 3H 2 0. As long as any pentahydrate remains, a definite 
dissociation pressure of 47 mm. of water vapour persists over the 
solid. If the water vapour is removed, the pentahydrate will give up 
more water, and the equilibrium pressure will be re-established. If 
the dehydration still goes on, the pressure will remain constant at 
47 mm. until all the pentahydrate has been converted into trihydrate, 
when the pressure will suddenly fall to 30 mm. This new pressure 
is the dissociation pressure of the trihydrate, which now begins to lose 
water and pass into the monohydrate CuS0 4 , H 2 0. As long as there is 
trihydrate present the pressure of 30 mm. will be maintained, but as 
soon as all the trihydrate has passed into monohydrate, the pressure 
again drops suddenly to 4*5 mm., which is the dissociation pressure of 
the latter substance. The solid dissociation product is now the 
anhydrous salt, and when all the monohydrate has been converted 
into this the pressure of water vapour drops to zero. 

What here holds good for the dehydration of hydrates also applies 
to the removal of ammonia from such compounds as AgCl , 3NH 3 , in 
which the ammonia may be removed in successive stages, with 
formation of intermediate compounds, e.g. 2AgCl , 3NH 3 . 

Sometimes dissociation results in the formation of two gaseous 

substances derived from one solid, e.g. the dissociation of ammonium 

salts, the solid chloride furnishing both ammonia and hydrochloric 

acid. In this case also there is a constant dissociation pressure for 

each temperature. If P is the constant vapour pressure of the 

undissociated ammonium chloride, and p the gaseous pressure of the 

ammonia or of the hydrochloric acid, these being equal since the two 

gases are produced in molecular proportions by the dissociation of the 

ammonium chloride, we have 

kP = k'p 2 , or p 2 - — = constant, 

whence the total dissociation pressure 2p is also constant. Balanced 
actions of this kind have been studied with ammonium hydrosulphide 
and similar compounds which dissociate at comparatively low 
temperatures. From the above equation it is apparent that the 
product of the pressures of ammonia and the acid is constant, Le, for 
the balanced action 

NH 4 HS % NH 3 + H 2 S 

the constant quantity is the product of the pressures of ammonia and 
hydrogen sulphide. If these substances are derived entirely from the 
dissociation of ammonium hydrosulphide, the pressures will be equal, 
but it is possible to add excess of one or other gas from the beginning, 


in which case they will no longer be equal. Their product, however, 
will retain the same value as before, though their sum, i.e. the total 
gas pressure, will have changed. This is evident from the equation 

JcP = h'pp'y or pp = — = constant, 


in which p and p f represent the pressures of the ammonia and the 
sulphuretted hydrogen respectively. Since p and p' enter into the 
equation in the same way, an excess of the one will have precisely 
the same effect on the equilibrium as an excess of the other. This 
conclusion also is verified by experiment, as may be seen from the 
following table, which contains some of the results of Isambert with 
ammonium hydrosulphide. Under p is the pressure of ammonia in 
centimetres of mercury, under p' the pressure of hydrogen sulphide, 
the other columns giving the variable total pressure and the constant 
product of the individual pressures. For this series the temperature 
of experiment was 17*3°. 







30 '0 


Excess of HaS-f 1 ^ 





Excess of NH 3 j^;7 





It is true that the numbers in the last column do not exhibit very 
great constancy, but this is due to experimental error, as comparison 
with other similar series of numbers serves to show. 

In the previous cases of balanced action we have had gaseous 
substances on one side of the chemical equation only. We now 
proceed to deal with a case where gaseous substances occur on both 
sides. If steam is passed over red-hot iron, a ferroso-ferric oxide of 
the composition Fe 4 5 is formed, the water being reduced to hydrogen. 
For the sake of simplicity we may assume that the product of oxida- 
tion of the iron is ferrous oxide, FeO, so that we have the equation 

H 2 + Fe = FeO + H 2 . 

If we continue to pass steam over the solid, all the iron is eventually 
oxidised, notwithstanding which, however, the action is a balanced one. 
For if we take the oxide produced, and heat it in a current of 
hydrogen, the hydrogen is oxidised to water, and the iron oxide 
reduced to metallic iron, the reduction being complete if the current 
of hydrogen is continued for a sufficient length of time. The reason 
why we in each case have the action completed is that the passage of 
the current of gas disturbs the equilibrium, which is never completely 
established. The nature of the equilibrium may be seen from the 

H 2 + Fe:>FeO + H 2 . 


On each side we have one solid substance and one gas. Since the 
gases are now on opposite sides of the equation, the equilibrium is 
regulated by the quotient of their pressures instead of by their product, 
as in the preceding case. Let p, P, p', F be the equilibrium pressures 
of the reacting substances in the order in which they occur in the 
chemical equation. We obtain the equilibrium equation 

p k'F 
kpP = k'v'F. or ^7 = -prr = constant. 
p JcP 

The ratio of the pressures of water vapour and hydrogen thus determines 
the equilibrium. If, as occurs when we pass water vapour over the 
iron, the ratio of the pressure of water to the pressure of hydrogen is 
greater than the equilibrium ratio, the action proceeds until all the 
iron has been converted into oxide. On the other hand, when we pass 
a current of hydrogen over the heated oxide, the above ratio is always 
less than the equilibrium ratio, and the oxide is completely reduced. 
The ratio changes with temperature, and becomes equal to unity at 
about 1000°. At this temperature, therefore, if we take equal volumes 
of water vapour and hydrogen, and pass the mixture over either iron 
or the oxide Fe 4 5 , no chemical action will take place, for the pressures 
are then the equilibrium pressures. 

Corresponding to the preceding cases of equilibrium with gases 
we have similar instances with substances in solution. In aqueous 
solution, however, the equilibrium is very often complicated by the 
occurrence of electrolytic dissociation. Several cases of this kind will 
be considered in a subsequent chapter. 

A solution equilibrium analogous to the dissociation of calcium 
carbonate is to be found in the action of water on insoluble salts of 
very weak insoluble bases combined with soluble acids. An organic 
base such as diphenylamine is so weak that its salts when dissolved 
in water split up almost entirely into the free acid and free base. 
Diphenylamine itself is practically insoluble in water, and so is the 
picrate formed from it by its union with picric acid. When the solid 
picrate is brought into contact with water, it partially dissociates with 
formation of insoluble base and soluble picric acid. We have, there- 
fore, the equation 

Diphenylamine picrate ^ Diphenylamine + Picric acid 

exactly analogous to the equation 

Calcium carbonate ^1 Calcium oxide + Carbon dioxide, 

with the exception that the picric acid is in the dissolved state, 
whereas the carbon dioxide is in the gaseous state. Now in the 
gaseous equilibrium we found that for each temperature a certain 
pressure, or concentration, of gas was produced. We should expect, 


therefore, that in the solution equilibrium for each temperature there 
should exist a certain osmotic pressure, or concentration, of the dis- 
solved substance which should be necessary and sufficient to determine 
the equilibrium, independent of the proportions in which the insoluble 
solid substances may be present. This has been confirmed by experi- 
ment. At 40*6° a solution of picric acid containing 13*8 g. per 
litre is in equilibrium with diphenylamine and its picrate, either 
singly or mixed in any proportions, and if diphenylamine picrate is 
brought into contact with water at this temperature it dissociates until 
the concentration of the picric acid in the water has reached this point. 

If we bring carbon dioxide at less than the dissociation pressure 
into contact with calcium oxide, no carbonate is formed. Similarly, 
if we bring at 40*6° a solution of picric acid, having a concentration of 
less than 13*8 g. per litre, into contact with diphenylamine, no 
diphenylamine picrate will be formed. On the other hand, if the 
solution has a greater concentration than 13*8 g. per litre, diphenyl- 
amine picrate will be produced until the concentration has sunk to 
that necessary for the equilibrium. This may be readily shown 
experimentally, on account of the different colours of diphenylamine 
and its picrate. The base itself is colourless, picric acid affords a 
yellow solution, and diphenylamine picrate has a deep chocolate- 
brown colour. A solution of picric acid at 40*6° containing 14 g. 
per litre at once stains diphenylamine deep brown, but a solution at 
the same temperature containing 13 g. per litre leaves the diphenyl- 
amine unaffected. 

If the weak base were soluble instead of insoluble, we should 
have an equilibrium for solutions corresponding to the dissociation of 
ammonium hydrosulphide. Urea is such a base, and the picrate of 
urea is very sparingly soluble as such in cold water, so that we have 
the equation 

Picrate of urea 1^. Urea + Picric acid. 

On the left-hand side of the equation we have a solid ; on the right- 
hand side we have two substances in solution. The sparingly soluble 
urea nitrate and oxalate afford similar instances. 

Phenanthrene picrate, when dissolved in absolute alcohol, dissociates 
into phenanthrene and picric acid, both of which are soluble. This 
case has been investigated, and found to obey the law of mass action. 
The calculation is, however, complicated on account of the picric acid 
being partially dissociated electrolytically, and part of the phenan- 
threne being associated in solution to larger molecules than that 
corresponding to the ordinary molecular formula. 

There are plenty of instances in solution corresponding to the 
action of steam on metallic iron. If barium sulphate is boiled with a 
solution of sodium carbonate, it is partially decomposed, according 
to the equation 


BaS0 4 + Na 2 C0 3 = BaC0 3 + Na 2 S0 4 . 

Here both the barium salts are insoluble and the sodium salts soluble, 
so that there is one solid and one salt in the dissolved state on each 
side of the equation. The active masses of the barium salts may be 
accounted constant during the reaction, for although they are gener- 
ally spoken of as "insoluble," they are in reality measurably soluble in 
water, cp. p. 307. The aqueous liquid in contact with them will therefore 
be and remain saturated with respect to them, i.e. their concentration 
and active mass in the solution will be constant. The equilibrium 
will thus be determined by a certain ratio of the concentrations of the 
soluble sodium salts, independent of what the actual values of the 
concentrations may be. Guldberg and Waage found by actual 
experiment that the concentration of the carbonate should be about 
five times that of the sulphate if the solution is to be in equilibrium 
with the insoluble barium salts. Such a solution will neither convert 
sulphate into carbonate nor carbonate into sulphate. If the proportion 
of carbonate in the solution is greater than 5 molecules to 1, the 
solution will convert barium sulphate into barium carbonate ; if the 
proportion is less than this molecular ratio, the solution will convert 
barium carbonate into barium sulphate. 

It should be mentioned that when we are dealing with salts in 
solution, Guldberg and Waage's Law has only an approximate applica- 
tion if we use concentration in the ordinary sense as the measure of 
the active mass of the dissolved substances, for by doing so we entirely 
neglect the effect of electrolytic dissociation. When the reactions are 
considered more closely, we find that it is usually the ions that are 
active, so that we should really deal with ionic concentrations in the 
majority of cases, instead of with the concentrations of the substance 
supposed undissociated. It happens, however, that in a great many 
cases the effect of dissociation is such that it affects the two opposed 
reactions equally, and in such cases the approximate conditions of 
equilibrium may be arrived at although dissociation is entirely 
neglected. In the instance given above of the equilibrium between 
soluble carbonates and sulphates and the corresponding insoluble 
barium salts, all the substances are highly dissociated, and approxi- 
mately to the same extent on both sides of the equation, so here the 
application of Guldberg and Waage's Law in its simple form gives 
results in accordance with experiment. 

If we increase the active mass of any substance playing a part in 
a chemical equilibrium, the balance will be disturbed, and the system 
will adjust itself to a new position of equilibrium, that action 
taking place in virtue of which the active mass of the substance 
added will diminish. Thus if we have carbon dioxide at a pressure 
such that the gas is in equilibrium with a mixture of calcium oxide 
and calcium carbonate, and suddenly increase its active mass by 


increasing the pressure upon it, a new equilibrium will be established 
by the action 

CaO + C0 2 = CaC0 3 

taking place, the tendency of which is to diminish the active mass, or 
pressure, of the carbon dioxide. Again, if we have an aqueous solution 
containing sodium sulphate and sodium carbonate in such proportions 
as to be in equilibrium with the corresponding barium salts, and add 
an extra quantity of sodium sulphate to the solution so as to increase 
the active mass of this salt, the equilibrium will be disturbed, and 
will readjust itself by means of the action 

Na 2 S0 4 + BaC0 3 = Na 2 C0 3 + BaS0 4 , 

which will go on until the concentration of the sodium sulphate has 
fallen to a value which gives the equilibrium ratio with the new con- 
centration of sodium carbonate. 

The principle here enunciated is of special importance in its 
application to cases of dissociation, whether gaseous or electrolytic. 
For convenience sake it may be stated for this purpose in the follow- 
ing form. If to a dissociated substance we add one or more of the 
products of dissociation, the degree of dissociation is diminished. 
Thus phosphorus pentachloride, which when vaporised dissociates 
according to the equation 

PC1 5 t pci 3 + ci 2 , . 

gives a vapour density little more than half that which corresponds 
to its usual formula. If, however, the pentachloride is vaporised in 
a space already containing a considerable quantity of phosphorus 
trichloride, the vapour density is such as would correspond to the 
formula PC1 5 for the pentachloride. Here one of the products of 
dissociation has been added, viz. PC1 3 , and the dissociation of the 
pentachloride has been in consequence to such an extent that the 
vapour density has practically the normal value. 

In the case of the dissociation of ammonium hydrosulphide, the 
addition of either ammonia or hydrogen sulphide to the normal 
dissociation products will diminish the degree of dissociation. Here, 
however, the undissociated substance exists only to a very small 
extent in the vaporous state, the consequence being that a smaller 
quantity of the salt is vaporised. This may be seen by reference to 
the numbers given on p. 246. At 17*3° the dissociation pressure is 
30 cm., half of this pressure being due to ammonia and half to hydrogen 
sulphide. If, now, we allow the hydrosulphide to dissociate into an 
atmosphere of hydrogen sulphide having a pressure of 36*6 cm., the 
increase in pressure will only be 10*7 cm., i.e. this will be the 
dissociation pressure under these conditions. If the dissociation is 
allowed to take place in an atmosphere of ammonia of 36 cm. pressure, 


we have a similar diminution of the dissociation to about one-third of 
its normal value. 

Suppose two substances are dissociating in the same space, and 
have a common product of dissociation ; it is evident from what has 
been said that the degree of dissociation of each will be lower than 
the value it would possess were the substances dissociating into the 
same space singly. Take, for example, a mixture of ammonium 
hydrosulphide and ethylammonium hydrosulphide. These substances 
dissociate according to the equations 

NH 3 (C 2 H 5 )HS t NH 2 (C 2 H 5 ) + H 2 S, 
NH 4 HS ^ NH 3 + H 2 S, 

and have hydrogen sulphide as a common dissociation product. 
When they dissociate into the same space, the effect is that as each 
supplies an atmosphere of hydrogen sulphide for the other to dissociate 
into, the dissociation pressure of each is diminished below the value 
it would have were it dissociating alone. Thus at 26*3° ammonium 
hydrosulphide has a dissociation pressure of 53*6 cm., and ethyl- 
ammonium hydrosulphide has a dissociation pressure of 13*5 cm. 
If the two substances did not affect each other when dissociating 
into the same space, the dissociation pressure of the mixture would 
be the sum of the dissociation pressures of the components of the 
mixture, viz. 67*1 cm. The value actually found by experiment is 
much lower than this, viz. 52*4 cm., which is even lower than the 
dissociation pressure of ammonium hydrosulphide itself. It should 
be stated that this result is not in accordance with the law of mass 
action, if pressures are taken as the measure of the active mass of the 
gases. The theoretical result is 55*3 cm., somewhat greater than the 
dissociation pressure of the ammonium hydrosulphide. 

When, as in the above instance, two unequally dissociated substances 
are brought into the same space, the more dissociated substance has 
a much greater effect on the degree of dissociation of the less dis- 
sociated substance than vice versa. This we might expect, for the more 
dissociated substance increases the concentration of the common dis- 
sociation product far above the value which would result from the 
dissociation of the less dissociated substance; while the latter, even 
if dissociated to its normal extent, would only slightly increase the 
concentration of the common dissociation product beyond the value 
obtained from the more dissociated substance alone. There is thus 
a great relative increase in the first case and a small relative increase 
in the second, the effects on the dissociation being in a corresponding 

It remains now to discuss the effect of temperature on balanced 
action. A rise of temperature is almost invariably accompanied by 
acceleration of chemical action. In a balanced action, therefore, both 


the direct and reversed actions are accelerated when the temperature 
is raised. The effect on the opposed reactions is, however, not in 
general equally great, with the result that the point of equilibrium 
is displaced in one or other sense. This displacement is intimately 
connected with the heat evolved in the reaction. If the direct action 
gives out a certain number of calories per gram molecule transformed, 
the reverse reaction will absorb an exactly equal amount of heat. 
Now rise of temperature always affects the equilibrium in such a 
manner that the displacement takes place in the direction which will 
determine absorption of heat. If, therefore, the direct action is accom- 
panied by evolution of heat, the action will not proceed so far at a 
high as at a low temperature, for if we start with equilibrium at the 
lower temperature, and then heat the system to a higher temperature, 
the heat-absorbing reverse action occurs, and the point of equilibrium 
moves backwards. If, on the other hand, the direct action absorbs 
heat, the action will proceed farther at high than at a low temperature. 
In all cases of gaseous dissociation at moderate temperatures the 
dissociation is accompanied by absorption of heat, so that the degree 
of dissociation increases as the temperature is raised. Thus the dis- 
sociation pressure of calcium carbonate rises with rise of temperature, 
and so does the dissociation pressure of salts like ammonium hydro- 
sulphide, as the following table shows : — 

Dissociation Pkessttke of Ammonium Hydrostjlphide 


Mm. of Mercury. 











Nitrogen peroxide, again, which at the ordinary temperature is only 
about 20 per cent dissociated into the simple molecules N0 2 , is at 
130° practically entirely dissociated. 

Since the heat of dissociation into ions is sometimes positive, 
sometimes negative, a rise of temperature may in some cases be 
accompanied by increased dissociation, in other cases by diminished 
dissociation. A considerable diminution of the degree of dissociation 
with rise of temperature has been proved for hydrofluoric acid and 
hypophosphorous acid. 

The rule which has here been applied to the displacement of 
chemical equilibrium with change of temperature is equally applicable 
to physical equilibrium. If we take a quantity of a liquid, and 
enclose it in a space greater than its own volume, a certain proportion 
of the liquid. will assume the vaporous state, heat being absorbed in 
the vaporisation. If now we raise the temperature, keeping the 
volume constant, the equilibrium will be disturbed, and the endo- 


thermic action will occur, i.e. more liquid will be converted into 
vapour, and the vapour pressure thus become greater. 

A number of cases of balanced action of an interesting type have 
been recently classified under the name of dynamic isomerism. 
The two substances, ammonium thiocyanate and thiourea, are isomeric, 
both having the empirical formula CSN 2 H 4 , and their isomerism at 
temperatures below 100° does not differ in any special way from the 
isomerism of other substances. If either substance is fused, however, 
it is partially transformed into the other isomeride, a balance being 
attained at a point where the liquid consists of about 80 per cent of 
ammonium thiocyanate and 20 per cent of thiourea. In this particular 
instance the substances have no marked tendency at the ordinary 
temperature to pass into each other when pure, so that the opposed 
reactions and the balance between them can be studied experimentally. 
Other instances have been investigated in which a change of one 
isomeride into the other in solution can be followed in the polarimeter, 
owing to a difference in the optical activity of the two substances. 
It is probable that most of the phenomena of " tautomerism " and 
"desmotropy" met with in organic chemistry are referable to similar 
causes. Liquids, for example, which are usually written with the group 

- CH 2 . CO - , very frequently act as if they contained the enol grouj), 

- CH : C(OH) - , and the liquids themselves often give values for their 
physical properties which would accord with their being mixtures of 
the two isomerides. That the liquids may be such mixtures is 
probable, seeing that other instances of isomeric balance are now well 

Cases of balanced action are discussed in the following papers which 
may be consulted by the student : — 

J. T. Cundall — " Dissociation of Nitrogen Peroxide " (Journal of the 
Chemical Society, lix. p. 1076 ; lxvii. p. 794). Compare also W. Ostwald, 
ibid. lxi. p. 242. 

J. Walker and J. R Applet ard — " Picric Acid and Diphenylamine ;j 
(ibid, lxix. p. 1341). 

J. Walker and J. S. Lumsden — " Dissociation of Alkylammonium 
Hydrosulphides " (ibid. lxxi. p. 428). 

T. M. Lowry — "Dynamic Isomerism" (ibid, lxxv. p. 235). 



In the preceding chapter we have had occasion to use the conception 
of reaction velocity in order to facilitate the discussion of chemical 
equilibrium, without entering into the question of how such a magni- 
tude can be practically determined. It is only in comparatively few 
cases that an accurate determination is possible at all, for the vast 
majority of reactions either take place so rapidly, or are complicated 
to such an extent by subsidiary reactions, that no specific coefficient 
of velocity for the reaction can be calculated. 

One of the simplest reactions, and one of the earliest to be studied 
with success, is the inversion of cane sugar. When this substance is 
warmed with a mineral acid in aqueous solution, it is gradually con- 
verted into a mixture of dextrose and levulose, and the process of 
conversion can be accurately followed by means of the polarimeter. 
The cane-sugar solution has originally a positive rotation, the invert 
sugar produced by the action of the acid has a negative rotation. If 
therefore we place the sugar solution in the observing tube of a polari- 
meter, and read off the angle of rotation from time to time, we can 
tell how the composition of the solution varies as time progresses 
without in any way disturbing the reacting system. For example, the 
original rotation of a cane-sugar solution was found to be 46*75°, whilst 
the rotation of the same solution after complete inversion was - 18*70°. 
The total change in rotation, therefore, corresponding to complete 
conversion of the cane sugar into invert sugar was 46*75° + 18*70° = 
65*45°. After the lapse of an hour from the beginning of the 
reaction, the rotation was found to be 35*75°. In that time the 
rotation had thus diminished by 11*00°, so that the fraction of the 
original sugar transformed was 11*00 -*- 65*45 = 0*168. By a similar 
calculation the quantity of sugar transformed at any other time could 
be arrived at. 

The following table gives the rotations actually observed at different 
times : — 


Time in Minutes. 


























- 7-00 







From this table it is evident that the rate at which the reaction 
proceeds falls off as less and less cane sugar remains in solution. In 
the first two hours the change in rotation is 20*75° ; in the two hours 
from 510' to 630' the change is only 3*00°. This is in accordance 
with the principle of Guldberg and Waage that the amount transformed 
in a given time will fall off as less cane sugar remains to undergo 
transformation. The chemical equation expressing the reaction is 

C^H^On + H 9 = C 6 H 12 6 + C 6 H 12 6 . 

(Dextrose) (Levulose) 

As the action progresses, water disappears as well as cane sugar; but if 
we consider that the action, takes place in aqueous solution, it is 
obvious that the change in the active mass of the water is very slight, 
and so for practical purposes of calculation, the active mass of the 
water may be taken as constant. The action is not a balanced one, 
but proceeds until all the cane sugar has been transformed. If A is 
the original concentration of the cane sugar, and x the quantity trans- 
formed at the time t, the rate of transformation at that time will be, 
according to the unimolecular formula, 

where dx represents the very small quantity transformed in the very 
small interval dt starting at the time t, and h is the coefficient of 
velocity of the action. From this equation the integral calculus 
enables us at once to find a relation between x and t } the corresponding 
values for any stage of the reaction in terms of the original concentra- 
tion and the velocity constant. This relation has the form 

log -^— = 0*4343^, 

JL — X 

or - log -: = constant. 

t ° A -x 

1 A 
The values of the expression — log -z are given in the last column 

t A — X 


of the preceding table, and it will be seen that they remain tolerably 
constant throughout the reaction. The agreement between experiment 
and the theory of Guldberg and Waage is therefore in this case quite 
satisfactory. 1 

The part played by the acid in the inversion of cane sugar is not 
well understood. The acid itself remains unaffected, but the rate of 
the inversion is nearly proportional to the amount of acid present if 
we always use the same acid. In such an action as this, the acid is 
said to act as an accelerator, and this behaviour on the part of acids is 
by no means uncommon, although the different acids vary very much 
in their accelerative power. As we shall see later, the accelerative power 
of acids furnishes us with a convenient measure of their relative 

A similar accelerating action of acids is found in the catalysis of 
ethereal salts. If a salt such as ethyl acetate is mixed with water, 
the two substances interact, with formation of acetic acid and ethyl 
alcohol (cp. Chap. XXII). This is in reality a balanced action, but 
if the solution of the ethereal salt is very dilute, the transformation is 
almost complete, and the action becomes of the same simple type as 
the inversion of cane sugar, the equation being 

CH 3 . COOC 2 H 5 + H 2 = CH 3 . COOH + C 2 H 5 OH. 

If no acid is present, the action takes place with extreme slowness, but 
in presence of the strong mineral acids, such as hydrochloric acid, it 
progresses with moderate rapidity. Although the progress of the 
action cannot conveniently be followed by a physical method, as in 
the sugar inversion, a chemical method may be employed without 
disturbing the reacting system. At stated intervals a measured 
portion of the solution is removed and titrated with dilute alkali. 
As the action progresses, the titre becomes greater, owing to the pro- 
duction of acetic acid, and from this increase we may deduce the 
amount of transformation. It is found that at each instant the rate 
of transformation is proportional to the amount of ethereal salt present, 
and a velocity constant may be calculated by means of the same 
formula as was used for sugar inversion. From a comparison of the 
velocity constants obtained with different acids, it appears that the 
relative accelerating influences of the acids is the same for both actions. 
Saponification of ethereal salts by alkalies affords us an example 
of a bimolecular reaction. The equation for the saponification of 
ethyl acetate by caustic potash is 

CH 3 . COOC 2 H 5 + KOH = CH 3 . COOK + C 2 H 5 OH, 
and the action proceeds until one or other of the reacting substances 

1 The inversion of cane sugar was studied, both practically and theoretically, by 
Wilhelmy, with the result arrived at above, before Guldberg and Waage enunciated their 
principle, and the numbers in the table are taken from Wilhelmy's work. 


entirely disappears. If we take both substances in equivalent propor- 
tions, and represent the active mass of each by a, the general equation 
for the rate of transformation will be 

integration of which leads to the equation 

t a(a - x) 

Experimental work confirms this conclusion, the expression on the 
right-hand side of the last equation being in reality constant. The 
course of the saponification can be easily followed by removing 
measured portions of the solution from time to time, and titrating 
them with acid. As the action proceeds, the amount of acid required 
to neutralise the potassium hydroxide in solution falls off proportion- 
ally. For a given ethereal salt, equivalent solutions of the caustic 
alkalies and the alkaline earths effect the saponification at practically 
the same rate, the rate of saponification for ammonia being very much 
smaller. For a given base the rate of saponification is greatly affected 
by the nature of both the acid radical and the alkyl radical which go 
to form the ethereal salt. Thus at 9*4° the velocity constants of 
various acetates on saponification by caustic soda are as follows : — 

Methyl acetate 3*493 

Ethyl acetate 2 '307 

Propyl acetate .1 '920 

Isobutyl acetate 1*618 

Isoamyl acetate 1*645 

While the corresponding numbers for various ethyl salts with the 
same base at 14*4° are — 

Ethyl acetate 3 '204 

Ethyl propionate 2*186 

Ethyl butyrate 1*702 

Ethyl isobutyrate 1*731 

Ethyl isovalerate . . . . . 0*614 

Ethyl benzoate 0*830 

Other instances of bimolecular reactions which have been investigated 
are the formation of sodium glycollate from sodium chloracetate and 
caustic soda, according to the equation 

CH 2 C1 . COONa + NaOH = CH 2 (OH) . COONa + NaCl, 

and the formation of such salts as tetra-ethyl ammonium iodide from 
tri-ethylamine and ethyl iodide, in accordance with the equation 

N(C 2 H 6 ) 3 + C 2 H 5 I = N(C 2 H 5 ) 4 I. 

A bimolecular reaction, which is, strictly speaking, a balanced action, 



but proceeds very nearly to an end in aqueous solution, is the forma- 
tion of urea from ammonium cyanate. According to the ordinary 

NH 4 CNO = CO(NH 2 ) 2 , 

we should expect the action to be unimolecular, but experiment has 
shown that it is a bimolecular reaction in dilute solutions, the reacting 
substances being the ions of the ammonium cyanate, with the 

NH' 4 + CNO'=CO(NH 2 ) 2 . 

In decinormal solution at ordinary temperatures, about 95 per cent 
of the ammonium cyanate is converted into urea. 

Trimolecular reactions are, comparatively speaking, rare, 
amongst those which have been most thoroughly investigated being 
the reduction of ferric chloride by stannous chloride, viz. 

2FeCl 3 + SnCl 2 = 2FeCl 2 + SnCl 4 , 

and the reduction of a silver salt in dilute solution by a formate, 
according to the equation 

2 AgC 2 H 3 2 + HC0 2 Na - 2Ag + C0 2 + HC 2 H 3 2 + NaC 2 H 3 2 . 

For such reactions we have the following expression for the rate : — 

where the reacting substances have each the original active mass a. 
On integration this becomes 

7 _ 1 . #(2a - x) 
t' 2a 2 (a-xf i 

and it has been ascertained experimentally that the expression on the 
right-hand side of the equation is actually a constant. 

No action has hitherto been investigated which follows the equation 
expressing the rate for a higher number of molecules than three. At 
first sight this is surprising, for in our ordinary chemical equations 
we are familiar with well-known reactions where the number of re- 
acting molecules is much greater than three. It would appear, how- 
ever, that these complicated reactions take place in stages, so that 
each is really an action composed of successive simple reactions. To 
take a simple instance, hydrogen arsenide is decomposed by heat into 
hydrogen and arsenic vapour. From the vapour density of arsenic it 
is known that the arsenic molecule contains four atoms under the 
conditions of experiment. We therefore write the equation for the 
decomposition as follows : — 

4AsH 3 = As 4 + 6H 2 . 


A study of the rate of the reaction, however, shows that it is not 
quadrimolecular, as the equation would lead us to suppose, but that 
it gives numbers agreeing with the unimolecular formula. This is 
evident from the following table, which gives the values of the 

expression k = - log , characteristic of unimolecular reactions, for 

t ex — x 

different times : — 

t is t k 

3 hours 0*0908 

4 „ 0*0905 

5 ,, 0*0908 

6 hours 0*0905 

7 „ 0*0906 

8 ,, 0*0906 

The action whose rate we measure, therefore, is a unimolecular action, 
most probably 

AsH 4 = As + 4H, 

which is then followed by the actions 

4 As = As 4 
2H = H 2 . 

Now, in order that the course of the total action may appear as a 
unimolecular action, it is a necessary assumption that the rate of the 
first action given above is much smaller than the rate of the succeeding 
actions. This is so because, in the first place, it is the slowest of a 
series of actions which will principally determine the rate from the 
initial to the final stage, and in the second place, if the total rate is to 
coincide with the rate of this slowest action, the rate of the others 
must be incomparably greater than this. An analogy may serve to 
"make the point clear. The time occupied in the transmission of a 
telegraphic message depends both on the rate of transmission along 
the conducting wire, and on the rate of the messenger who delivers 
the telegram; but it is obviously this last, slower, rate that is of 
really practical importance in determining the total time of trans- 
mission, and, indeed, the speed of electricity may be neglected alto- 
gether in the calculation. 

When we measure the rate of a complicated action, then, we are 
in general measuring an average rate of a series of reactions which 
may be progressing successively or simultaneously, and it is the rate 
of the slowest of these actions which plays the principal part in 
determining the total rate. Should the other actions proceed at an 
immeasurably faster rate than the slowest action, the whole action 
will appear to go at a rate and be of a type regulated by this action 
alone. Hence it is that we so frequently find complex actions pro- 
ceeding in such a way as to suggest that they are much simpler in 
type than the total action really is. 

The saponification of an ethereal salt of a bibasic acid affords a 


good instance of the progress of a reaction in stages. For example, 
the saponification of diethyl succinate by caustic soda has been shown 
to proceed, not according to the equation 

C 2 H 4 (COOC 2 H 5 ) 2 + 2NaOH = C 2 H 4 (COONa) 2 + 2C 2 H 5 OH, 

but according to the equations 

C 2 H 4 (COOC 2 H 5 ) 2 + NaOH = C 2 H 4 (COOC 2 H 5 )(COONa) + C 2 H 5 OH, 
C 2 H 4 (COOC 2 H 5 )(COONa) + NaOH = C 2 H 4 (COONa) 2 + C 2 H 5 OH. 

On the last assumption, Guldberg and Waage's principle leads to a 
certain expression for the velocities of the total action involving the 
two velocity constants of the single actions, and the experimental rate 
has been found to agree with the theoretical requirements. The fact 
that the action does in reality proceed in two stages may be easily 
demonstrated by treating ethyl succinate in alcoholic solution with 
half the calculated quantity of caustic potash necessary for complete 
saponification. Instead of half the ethereal salt being completely saponi- 
fied, and half being left untouched, about three-fourths of the original 
quantity is, under favourable conditions, converted into the potassium 
ethyl salt C 2 H 4 (COOEt)(COOK), the product of the first stage, one- 
eighth remaining unattacked, and one -eighth being converted into 
the dipotassium salt. 

In general, then, it may be accepted as a fact that in actions which 
are expressed by comparatively complicated chemical equations, in- 
volving the interaction of a large number of molecules, we are dealing 
with a series of simpler actions, not more than three molecules being 
involved in each of these. 

The velocity of a given action is usually greatly affected by 
Change Of temperature and change of the medium in which the action 
occurs. Almost invariably a rise of temperature is accompanied by a 
large increase in the rate of the reaction, the speed being very 
frequently doubled for a rise of five or ten degrees starting at the ordin- 
ary temperature. It is somewhat difficult to account for this very high 
temperature coefficient. On any of the usual hypotheses regarding 
the rate of molecular motion and its variation with the temperature, 
it is impossible to assume that the speed of the molecules increases so 
greatly that they encounter each other twice as often when the 
temperature rises from 15° to 20°. It has been suggested that 
only a certain small proportion of the total number of molecules are 
active at any one temperature, and that this number increases rapidly 
as the temperature rises. On this supposition ions might be supposed 
to be almost all in the active state, at least so far as double 
decompositions amongst acids, salts, and bases are concerned, for these 
actions progress so rapidly that their speed has never been measured. 
An action which lends some support to this supposition is that of very 




dilute acids on zinc, which progresses at a rate which is almost 
independent of the temperature. This action is, in terms of the 
dissociation theory, 

Zn + 2H' = Zn" + H 2 

The solid zinc can scarcely have its active part greatly affected by 
change of temperature (since at most only the superficial part of it 
can take part in the action), and the hydrogen ions must be considered 
nearly all active at the ordinary temperature, as a rise of tempera- 
ture does not increase the rate of the reaction. 

Very slight changes in the nature of the medium also greatly affect 
the speed of a reaction. Thus if we remove 1 per cent of the water in 
which the conversion of ammonium cyanate into urea is taking place, 
and bring the solution up to its original volume by adding acetone, 
which takes no part in the reaction, the rate of the conversion increases 
by nearly 50 per cent. In ethyl alcohol the rate of transformation 
of the cyanate into urea is thirty times as great as it is in pure water, 
other conditions remaining the same, notwithstanding the fact that the 
number of ions in the alcoholic solution is much smaller than the 
number in pure water of the same concentration (cp. p. 223). 

The following table contains the coefficients of velocity observed 
for the bimolecular reaction 

N(C 2 H 6 ) 3 + C 2 H 5 I = N(C. 2 H 5 ) 4 I 

in different solvents. To avoid unnecessary ciphers, the actual 
numbers have been multiplied by 1000. 



Hexane . . . . . . 0*180 

Heptane . 




Benzene . 


Ethyl acetate 


Ethyl ether 


Methyl alcohol 


Ethyl alcohol 


Allyl alcohol 


Benzyl alcohol 


Acetone . 


The great range of the speed of one action is very well shown by this 
table. We cannot imagine that the reacting molecules meet each 
other in benzyl alcohol 700 times as often in a given time as they do 
in hexane. Either we must assume that there is a greater proportion 
of active molecules in the former solvent than in the latter, or that 
the addition takes place at a larger proportion of the encounters of the 
reacting molecules, a supposition which is practically identical with 
the first. 

We very frequently find that below a certain temperature chemical 
action apparently will not occur, while above that temperature the 


action takes place freely. Thus we speak of the temperature of 
ignition of a mixture of gases, meaning usually thereby the lowest 
temperature which, when given to one part of the mixture, will produce 
chemical action in the whole mass. For example, we say that the 
temperature of ignition of a mixture of air and saturated carbon 
bisulphide vapour is about 160°, because a glass rod heated to that 
temperature and applied to any part of the mixture will inflame the 
whole. If we inquire more closely into such actions, however, we 
generally find that the action occurs at temperatures below the ignition 
temperature, but that it will not then propagate itself under ordinary 
circumstances. The experimental proof of this is given by maintaining 
the whole mixture at a temperature somewhat below the ignition point, 
and noting after a time if the action has progressed. The reason for 
the non- propagation at lower temperatures is that the action then 
progresses only slowly. The heat evolution consequent on the 
chemical change is therefore spread over such an interval of time 
that the temperature of the mixture remains below the ignition 
temperature, the action becoming slower and slower if no external 
heat is supplied, eventually to cease. At the ignition temperature the 
action takes place at such a rate that the heat evolution is sufficiently 
rapid to keep the temperature of the gaseous mixture up to the ignition 
point, and even to raise it still higher, with the result that the action 
proceeds at an ever-increasing rate. We therefore see that the so-called 
temperature of ignition depends on the generally - observed rapid 
increase in the rate of chemical action with rise of temperature, and 
may vary with the original temperature of the whole mixture. 

The same thing holds good, and may be followed more easily, with 
certain solids. Solid ammonium cyanate, for instance, may be kept 
for many months at the ordinary temperature without undergoing any 
notable transformation into urea, according to the equation 

NH 4 CNO = CO(NH 2 ) 2 . 

At 60° the action is fairly rapid, if the temperature is kept up 
externally, but the heat evolution is still too slow to enable the action 
to go on at an increasing rate. If the external temperature is 80°, 
however, the action proceeds swiftly, and the rapid heat evolution 
raises the temperature to such an extent that the whole passes almost 
instantaneously into fused urea, the melting point of which is 132°. 

If the heat evolution is comparatively small, as it is with ammonium 
cyanate, the increasing rapidity of the reaction may be observed 
through a considerable range of temperature. If, on the other hand, 
the heat evolution is great, as it is with most explosives, the action 
when it takes place perceptibly is usually propagated at once, owing 
to the rapid rise in temperature of the particles in the immediate 
neighbourhood of the reacting particles. The rate of propagation of 
explosion in solid explosives when fired is very great, rising in some 


instances to about five miles per second. The rate of propagation of 
the explosion wave in gaseous mixtures, such as that of oxygen and 
hydrogen, is somewhat less than this, averaging about a mile and a half 
per second, and being therefore of the same dimensions as the average 
rectilinear velocity of the gaseous molecules at the temperature of the 
explosion (cp. p. 87). 

Substances which react vigorously at the ordinary temperature 
usually lose their chemical activity entirely when cooled to the tem- 
perature of boiling liquid air. This we must attribute to the lowering 
of the rate of chemical action by fall of temperature, the speed being 
so greatly diminished that the action does not perceptibly occur at all in 
any moderate length of time. That there is not an entire cessation of 
action is probable, since in certain cases we can follow the gradual slack- 
ening of the action to extinction as the temperature falls. Thus sodium 
and alcohol, which react briskly at the ordinary temperature, with evolu- 
tion of hydrogen, become less and less active as the temperature is 
lowered, until finally the hydrogen formed by the action is so small 
in quantity as to escape observation. If we reflect that we often see 
the reaction velocity halved by a fall of 5° in temperature, we can 
conceive that a fall of 100° might by successive halving reduce the 
reaction velocity to a millionth of its original value. We may some- 
times find it convenient, in accordance with this, to look upon chemical 
inactivity as being not absolute inactivity, but rather the progress of a 
chemical action at a rate too slow for measurement, or even detection. 

We have now to consider briefly the points of resemblance between 
physical transformation and chemical transformation. In the change 
from solid to liquid, and vice versa, there is (for a given pressure) a 
definite temperature of transformation (p. 60). Above this tempera- 
ture the solid passes into the liquid; below this temperature the liquid 
passes into the solid. So it is for the transformation of one crystalline 
modification into another, as in the example of the two modifications of 
sulphur. Above the inversion temperature, one modification is stable ; 
below the inversion temperature, the other. If a liquid is brought to 
a temperature slightly below its inversion temperature (the freezing 
point) and brought into contact with the more stable crystalline phase, 
it will assume the crystalline state, the rate of crystallisation being for 
small temperature differences proportionate to the degree of overcooling. 
A maximum rate, however, is attained when the degree of overcooling 
reaches a certain value, and below the temperature corresponding to 
this, the rate of crystallisation rapidly falls with further diminution of 
temperature (cp. p. 62). The same thing has been observed with 
the reciprocal transformation of crystalline modifications : at first the 
rate of transformation increases as the temperature falls below the tem- 
perature of inversion, afterwards however to diminish rapidly as the 
fall of temperature proceeds. No doubt monoclinic sulphur, cooled to 
a temperature considerably below zero, would exhibit little tendency to 


pass into the more stable rhombic form, for even at ordinary tempera- 
tures the process takes a comparatively long time. We have already 
seen that an overcooled " glass " may remain for a very long time in 
contact with the more stable crystalline modification if the overcooling 
is sufficiently great, without crystallisation progressing at a sensible 
rate. If we heat the " glass," however, to a higher temperature, the 
transformation proceeds with ever-increasing rapidity until a tempera- 
ture a few degrees below the inversion point is reached. This is 
evidently comparable to the chemical transformation of solid ammonium 
cyanate into urea. Ammonium cyanate is the less stable form, and 
although it may be kept for a very long time at 0° in contact with urea 
without undergoing appreciable transformation, the transition goes on 
at an increasing rate as the temperature is raised. Here the inversion 
point is not known, but it must at least be considerably over 80°. The 
reverse transformation of urea into ammonium cyanate has only as yet 
been investigated in aqueous solution. 

When cyanic acid vapour condenses below 150°, it forms 
cyamelide ; when it condenses above 150°, it forms cyanuric acid; 
the chemical action being in both cases one of polymerisation, as 
cyanic acid has the formula CNOH, cyanuric acid the formula 
(CNOH) 3 , and cyamelide a still more complicated formula, expressed 
generally by (CNOH) n . This would indicate that cyanuric acid had 
the inversion point 150°, cyamelide being the stable form below that 
temperature, and cyanuric acid the stable form above that temperature. 
The conversion of cyamelide into cyanuric acid above 150° has actually 
been observed, but the reverse action has not hitherto been noticed, 
probably on account of its slowness. If we construct a pressure- 
temperature diagram for the three phases, we shall find it exactly 
analogous to the diagram for the three physical states of aggregation 
of water (Fig. 9, p. 64). 

From the foregoing instances it is evident that considerable analogy 
exists between the effect of temperature on chemical and on physical 
change, especially when the chemical change is reversible : and it may 
often be found expedient to look upon apparently irreversible chemical 
changes as being in reality reversible but with a temperature of inver- 
sion so high that the reverse action has never been realised. 

It will be noted that in all of the above instances there can be no 
coexistence in equilibrium of two systems which are mutually insoluble 
and capable of reciprocal transformation, except at the inversion point. 
Above the inversion point one of the systems is stable, below the in- 
version point the other is stable ; at the inversion temperature itself, 
both systems are equally stable. This rule holds good both for chemi- 
cal and physical changes. If, on the other hand, the chemical systems 
are mutually soluble, there can be equilibrium at any temperature for 
which they only form one phase, the proportions of each system present 
changing in this case with the temperature. The fused mixture of 


ammonium thiocyanate and urea forms an example of this one-phase 
equilibrium of two reciprocally transformable systems. If we fuse 
thiourea, a certain proportion of it passes into ammonium thiocyanate 
with a measurable velocity, and we should expect that a different pro- 
portion would be transformed according to the temperature at which 
the system was maintained. Another example of one-phase equilibrium 
of reciprocally transformable systems is to be found in the solutions 
of dynamic isomerides (p. 253), or in the substances themselves if 
they be liquid. The actual transformation in this case also has been 
noted, and its velocity measured. 

Traces of water vapour have been found to play a very important 
part in determining the occurrence, or at least the rate, of many 
chemical actions. Thus ammonia and hydrochloric acid, when both in 
the gaseous state, unite readily under ordinary circumstances to form 
ammonium chloride. If precautions are taken, however, to have the 
gases absolutely dry, they may be mixed without any union taking 
place. On the other hand, ammonium chloride, when vaporised, dis- 
sociates to a very great extent into ammonia and hydrochloric acid, 
as is rendered evident by the vapour density being only about half the 
normal value calculated from the molecular formula NH 4 C1 by the help 
of Avogadro's principle. If the ammonium chloride is perfectly dry, 
however, the vapour density is normal, thus showing that no dissocia- 
tion has taken place. The reversible action 

NH 3 + HC1 t- NH 4 C1 

is therefore apparently dependent on the presence of traces of water 
for its occurrence either in the direct or the reverse sense. No satis- 
factory explanation of the action of the moisture has yet been given. 
It has not indeed been clearly established whether the action is alto- 
gether inhibited by the absence of water vapour or whether it still 
goes on, but at a greatly diminished rate. In the latter case the action 
of the water might be comparable to the action of acids in accelerating 
the inversion of cane sugar, the hydrolysis of ethereal salts, and the like. 

The following papers dealing with rate of chemical action may be con- 
sulted : — 

HARCOURTand Esson (Philosophical Transactions, 1866, p. 193, and 1867, 
p. 117): Interaction of Permanganate with Oxalic Acid, and of Hydrogen 
Peroxide with Hydriodic Acid. 

R. Warder (American Chemical Journal, vol. iii. No. 5) : Saponifica- 
tion of Ethyl Acetate. 

A. A. Noyes and G. J. Cottle (ibid, vol. xxi. p. 250) : Reduction 
of Silver Acetate by Sodium Formate. 

J. Walker and others (Journal of the Chemical Society, lxvii. p. 489 ; 
lxix. p. 195 ; lxxi. p. 489): Transformation of Ammonium Cyanate into 



It is customary and correct to speak of sulphuric acid as a strong acid, 
and of acetic acid as a weak acid, and the statement is the outcome of 
our general experience of the chemical behaviour of these substances. 
In comparing two such acids there is no difficulty ; they are so different 
in their properties that no one could mistake their relative strengths. 
But if we compare two acids which are closer together in the scale of 
strength, say hydrochloric and nitric acids, it is impossible from our 
general chemical experience alone to say which is the stronger, and 
we must resort to a more exact definition of strength and to more 
accurate experiment. 

A method which, when properly applied, leads to useful and 
consistent results, is that of the displacement of one acid from its 
salts by another. If an equivalent of sulphuric acid be added to a 
given quantity of an acetate in solution, the change in properties of 
the solution is sufficient to indicate that practically the whole of the 
sulphuric acid has been neutralised, and the corresponding quantity of 
acetic acid liberated. There is therefore no doubt that the sulphuric 
acid is much stronger than the acetic acid, being capable of turning 
the latter out of its salts. But the method must be applied with 
caution, or it leads to contradictory and inconsistent results. If, for 
example, we take sodium silicate in aqueous solution and add hydro- 
chloric acid, sodium chloride will be formed and silicic acid liberated, 
but yet at a very high temperature, as in the process of glazing earthen- 
ware, silicic anhydride in presence of water vapour is capable of 
decomposing sodium chloride with expulsion of hydrochloric acid. 
Without any further principle than that of displacement to guide us, 
the first experiment would show that hydrochloric is stronger than 
silicic acid, and the second that silicic acid is stronger than hydro- 
chloric acid. Again, if we pour aqueous acetic acid on sodium 
carbonate, there is immediate effervescence due to the expulsion of 
carbonic acid. Yet, if we pass a stream of carbonic acid into a 
saturated aqueous solution of sodium acetate, a precipitate of sodium 


hydrogen carbonate soon makes its appearance, the acetic acid having 
been expelled from its salt by carbonic acid. Again, a solution of 
potassium acetate in nearly absolute alcohol is decomposed by carbonic 
acid to a great extent with precipitation of carbonate. From these 
experiments it is impossible to say whether acetic or carbonic acid is 
the stronger, since the expulsion takes place in different senses accord- 
ing to the conditions of experiment. There is, of course, no doubt 
among chemists that hydrochloric acid is much stronger than silicic acid, 
and that acetic acid is much stronger than carbonic acid. We must 
therefore inquire more closely into the experiments which seem to 
point to the contrary conclusions. In the first place, it is obvious 
from the experiments themselves that we are dealing with actions 
which can, according to circumstances, take place in either sense, i.e. 
with balanced actions. Carbonic acid, or its anhydride and water, 
can always displace a little acetic acid from its salts, although it is a 
much weaker acid. If all the substances remain within the sphere of 
the reaction this displacement will not go far, as the reverse reaction 
will set in and soon establish equilibrium. The action expressed by 
the equation 

CH 3 . COOK + C0 2 + H 2 = KHC0 3 + CH 3 . COOH 

would therefore speedily come to an end if the products of the action 
were to remain and accumulate in the system. But if one of the 
products, say potassium hydrogen carbonate, is insoluble, or nearly so, 
its active mass cannot increase beyond a certain small amount, however 
much of it may be formed, for it falls out of solution, i.e. the true 
sphere of action, as soon as it is produced. Whilst, therefore, in aqueous 
solution, in which all the substances remain dissolved, the carbonic 
acid succeeds in displacing only a small proportion of the acetic 
acid, in alcoholic solution it displaces a much larger amount, owing to 
the insolubility of one of the products of reaction. The comparatively 
great displacement in saturated sodium acetate solution is referable to 
the same cause, sodium hydrogen carbonate being very slightly soluble 
in a saturated solution of sodium acetate. 

In the case of the action of silica on a chloride, we have again one 
of the substances removed from the sphere of action as it is produced, 
viz. hydrochloric acid, which at the high temperature of the experi- 
ment escapes as vapour. In solution, only an infinitesimally small 
proportion of hydrochloric acid would be displaced by silicic acid, 
owing to the reverse reaction which would at once set in, but at 
the high temperature no reverse action is possible at all, since one 
of the reacting substances leaves the sphere of action as soon as it is 

Similarly, we are not in a position to judge of the relative strengths 
of hydrochloric and hydrosulphuric acids by experiments with sulphides 
insoluble in water. Sulphuretted hydrogen will at once expel hydro- 


chloric acid from copper chloride, even if excess of hydrochloric acid 
is present in solution. Yet sulphuretted hydrogen is a very feeble 
acid compared to hydrochloric acid. The displacement is due to the 
insolubility of the copper sulphide, which is removed from the sphere 
of action as it is produced. That such experiments lead to no definite 
conclusion may be seen by taking the same acids with another base. 
Though sulphuretted hydrogen is passed through a solution of ferrous i 
chloride until the solution is saturated with it, a scarcely percep- 
tible precipitate of ferrous sulphide will be formed, and if a little 
hydrochloric acid is added to the solution from the beginning, no * 
precipitate will be formed at all. Here, then, the same two acids 
exhibit totally different behaviour relatively to each other, according 
to the nature of the base for which they are competing. 

Now, in the above instances the acids have been selected of as 
widely different natures as possible, so that the fallacy of the reasoning 
based on the experiments mentioned is obvious. But similar fallacious 
reasoning is prevalent, and passes without detection, when experiments 
are discussed regarding acids where common chemical experience 
supplies no answer as to their relative strengths. Thus it is almost 
invariably a settled conviction in the minds of students that sulphuric 
acid is a stronger acid than hydrochloric acid because it expels the latter 
from its salts. No doubt this is the fact if we evaporate a solution of 
the chloride and sulphuric acid to dryness, or nearly so. But, of course, 
in this case the hydrochloric acid is expelled as vapour, and cannot 
therefore participate in the reverse reaction. The hydrochloric acid 
is expelled, not because it is a feebler acid than sulphuric acid, but 
because it is more volatile. 

These examples suffice to show that expulsion of one acid from its 
salts by another cannot be used as a proof that the latter is the stronger 
acid, unless the two acids are competing under circumstances equally 
favourable to both. The proper conditions are secured when the 
reacting systems form only one phase, namely, that of a solution. 
As soon as one of the components of the reacting systems is removed 
as a gas or as an insoluble solid or liquid, the system which does not 
contain that substance as one of its components is unduly favoured at 
the expense of the other system. 

If we are to compare the strengths of hydrochloric and sulphuric 
acids, then, we shall do best to take a soluble sulphate and add to it 
an equivalent of hydrochloric acid, the base being so chosen that the 
chloride formed is also soluble. Since all salts which have potash or 
soda as base are soluble, a potassium or sodium salt is generally selected, 
and the competing acid added to its aqueous solution. Thus equivalent 
solutions of hydrochloric acid and sodium sulphate may be mixed, and 
the composition of the resulting solution investigated, in order to 
ascertain in what proportion the base distributes itself between the 
two acids, the assumption being that the stronger acid takes the greater 


share of the base. In general, it is necessary to use a physical method 
for determining the composition of the solution, since the application 
of a chemical method would disturb the equilibrium. That an 
equilibrium is actually being dealt with is ascertainable from the 
fact that the solution has exactly the same properties in all respects, 
whether the base was originally combined with the sulphuric acid or 
with the hydrochloric acid, as will be presently shown in a numerical 
example. The two methods which have been most extensively applied 
are the thermochemical method of Thomsen and the volume method 
of Ostwald. 

When a gram molecule of sulphuric acid in fairly dilute solution 
(about one-fourth molecular normal) is neutralised by an equivalent 
amount of caustic soda of similar dilution, a production of 313*8 
centuple calories is observed. The same amount of soda neutralised by 
hydrochloric acid is attended by a heat evolution of 274*8 K. Now if 
the addition of hydrochloric acid to a solution of sodium sulphate pro- 
duced no effect chemically, we should also expect no thermal effect. If, 
on the other hand, all the sulphuric acid were expelled from combination 
with the soda, we should expect an absorption of 313*8 - 274*8 K — 
39 K. Now an actual heat absorption of 33*6 K per gram 
molecule was observed by Thomsen. This would indicate that the 
greater proportion of the base was taken by the hydrochloric acid, an 
equivalent quantity of sulphuric acid being expelled from its combina- 

■ tion with the soda. If we assumed that the heat absorption were 
directly proportional to the amount of chemical action, the proportion 
of sulphate converted into chloride would be 33*6 -*- 39 = 0*86. This 

* proportion, however, is not the correct one, for the sulphuric acid 
liberated reacts with the normal sodium sulphate remaining, to produce 
a certain quantity of sodium hydrogen sulphate, the formation of 

! which is attended by absorption of heat, so that the total heat absorp- 
tion observed is too great. Special experiments show that the correc- 
tion to be applied in order to eliminate the effect of this action is 7*6 K, 
- the heat absorption due to the displacement of sulphuric by hydro- 
chloric acid thus being 33*6 - 7*6 = 26 K. The proportion of sulphuric 
acid expelled is therefore 26-^39, or two-thirds. When, therefore, 
equivalent quantities of sulphuric and hydrochloric acids compete for a 
quantity of base sufficient to neutralise only one of them, the hydro- 
chloric acid takes two-thirds of the base and the sulphuric acid one- 
third. The reverse experiment of adding sulphuric acid to a solution 
of sodium chloride showed that the final distribution of the base 
between the acids was the same as above so far as could be judged 
from the heat effect. Since the hydrochloric acid always takes the 
larger share of the base, we conclude that it is the stronger acid, at 
least in aqueous solution. 

Thomsen, by working in this way, compiled a table of the avidi- 
ties of different acids, from which it is possible to tell at once how a 


base will distribute itself between any two of them if all three sub- 
stances are present in equivalent proportions. The avidities of some 
of the commoner acids are given below : — 

Acid. Avidity. 

Nitric 100 

Hydrochloric 100 

Sulphuric 49 

Oxalic 24 

Orthophosphoric 13 

Monochloracetic 9 

Tartaric 5 

Acetic 3 

In order to find the distribution ratio from this table, we proceed as 
follows. Let the acids be sulphuric and chloracetic, the avidities being 
49 and 9 respectively. If the base and these acids are present in 
equivalent proportions, the base will share itself between the acids in 
the ratio of their avidities, i.e. the sulphuric acid will take |~| and the 
chloracetic acid -g 9 ¥ . 

Ostwald's volume method is based on similar principles. Instead 
of heat changes, the changes of volume accompanying chemical reactions 
are measured. The substances used by him were contained in aqueous 
solutions of such a strength that a kilogram of solution contained 1 
gram equivalent of acid, salt, or base. The specific volumes of these 
solutions were carefully determined so that the change of volume pro- 
duced by chemical action might be ascertained. Thus the volume of a 
kilogram of potassium hydroxide solution was found to be 950*668 cc., 
and of a nitric acid solution 966 '623 cc. If, on mixing these solutions, 
no change of volume occurred, the total volume would be 1917*291 cc. 
But the volume actually found on mixing the solutions was 1937*338 cc. 
The neutralisation of the acid and base is thus accompanied by an 
expansion of 20*047 cc. Similarly, changes of volume accompany other 
chemical reactions, and the extent to which a given action has occurred 
can be measured by the volume change. A solution of copper nitrate 
had a volume equal to 3847*4 cc, and an equivalent solution of copper 
sulphate 3840*3 cc. Solutions of nitric and sulphuric acids had the 
volumes 1933*2 and 1936*8 respectively. If no action occurred on 
mixing the copper sulphate solution with the nitric acid solution, the 
total volume would be 3840*3 + 1933*2 = 5773*5 ; if complete trans- 
formation into copper nitrate and sulphuric acid took place, the total 
volume would be 3847*4 + 1936*8 = 5784*2. The actual volume 
found by mixing the copper nitrate and sulphuric acid solutions was 
5780*8, and by mixing the copper sulphate and nitric acid solutions 
5781*3. These two volumes are practically identical, and we may take 
as their mean 5781*0. We have therefore the numbers — 

All Copper Sulphate. Actual. No Copper Sulphate. 

5773*5 5781*0 5784'2 

Difference 7*5 3*2 


The actual equilibrium is evidently nearer the system containing no 
copper as sulphate than the system containing all the copper as 
sulphate, and if we assume direct proportionality, the base is shared by 
the nitric and sulphuric acids in the ratio of 7*5 to 3*2, or nitric acid 
takes 70 per cent of the base, leaving the sulphuric acid 30 per cent. 
This result is not quite accurate, as allowance has to be made for the 
slight volume changes consequent on the action of the respective acids 
on their neutral salts. "When this correction is applied, it appears that 
the nitric acid takes 60 per cent of the base and the sulphuric acid 40 
per cent. 

A table of avidities can be constructed for the different acids from 
similar data, and a comparison with Thomsen's avidities derived from 
I thermochemical experiments shows that the two methods yield results 
sin harmony with each other, at least so far as relative order of the 
■ acids is concerned. The actual avidity numbers differ considerably in 
many instances, but it has to be borne in mind that the thermochemi- 
cal measurements are on the whole much less accurate than the volume 
measurements, and the numbers derived from them consequently less 

In special cases the distribution of a base between two acids may 
be studied by making use of other physical properties than those 
already mentioned. For example, measurements of the refractive in- 
dex of solutions often lead to satisfactory results, and also measure- 
i ments of the rotatory power when optically active substances are in 
, question. The principle involved is identical with that just described, 
t any differences being merely differences in detail. 

A method which differs in principle from the distribution of a base 
between two competing acids, and may also be applied to the deter- 
mination of the relative strengths of acids, is to ascertain the accelerat- 
ing influence exerted by different acids on a given chemical action. 
For example, the inversion of cane sugar has long been known to take 
place much more rapidly in presence of an admittedly strong acid like 
sulphuric or hydrochloric acid, than it does in presence of an equiva- 
8 lent quantity of an admittedly weak acid like acetic acid. The strong 
mineral acids have thus a greater accelerating effect than the weak 
organic acids, and it is natural to infer that an exact determination of 
jl the specific accelerating powers of different acids might lead to a 

i knowledge of their relative strengths. There is no obvious connection 
between this method and the preceding method of relative displace- 
ment, but a connection exists, as will be shown later, and the results 
obtained are in general quite in harmony with each other. 

The method of sugar inversion as practised by Ostwald was 
performed in the following manner. Normal solutions of the various 
acids were mixed with an equal volume of 25 per cent sugar solution 
and placed in a thermostat whose temperature remained constant at 
25°. The rotation of each solution was taken from time to time, and 


a velocity constant calculated according to the formula given on p. 255. 
The order of these velocity constants is the measure of the accelerating 
powers of the acids, and presumably a measure of their relative strengths. 

Another action well adapted to investigating the accelerating power 
of acids is the catalysis of methyl acetate (cp. p. 256). Ostwald 
mixed 1 cc. of normal acid with 1 cc. of methyl acetate, and diluted the 
mixture to 15 cc. This solution was then placed in a thermostat at 
26°, and its composition ascertained at appropriate intervals in the 
manner already indicated. A calculation of the velocity constant by 
the usual formula for unimolecular reaction gave the required measure 
of the accelerating power. 

A comparison of the results obtained by the different methods is 
given in the following table, the value for hydrochloric acid being 
made in each case equal to 100, in order to assist the comparison : — 

Velocity Constants. 


Sugar Inversion. Catalysis of Acetate. 



100 100 



100 91-5 



53 547 



18'6 17'4 






4-8 4"3 






0*4 0-35 

It is at once evident that the order in which the acids follow each 
other is the same in all cases, and in especial it will be seen that the 
numbers expressing the accelerating powers of the acids are closely 
similar, although the accelerating influence was exerted on entirely 
different chemical actions. The avidity numbers differ considerably 
from the others, but the general parallelism of the results cannot be 
denied, and we are therefore justified in adopting the acceleration 
method as a means of measuring the relative strengths of acids, 
although its theoretical justification is not immediately obvious. 

It was pointed out by Arrhenius that if we arrange the acids in 
the order of their relative strengths, they are also arranged in the 
order of the electrical conductivities of their equivalent solutions. 
This may be seen in the following table, the first column of which 
contains the mean value of the velocity constants of sugar inversion 
and catalysis of methyl acetate, and the second that of the electric 
conductivities of equivalent solutions, all values being referred to that 
for hydrochloric acid as 100 : — 

Velocity Constants. 

Electric Conductivity. 

























The parallelism is here unmistakable, the numerical values in the two 
jolumns being often practically identical. 

At first sight it appears a matter of difficulty to associate the 
electric conductivity of an acid with its strength, i.e. its chemical 
activity in so far as it behaves as an acid ; but the dissociation 
lypothesis of Arrhenius furnishes the clue to the nature of the con- 
nection. All acids in aqueous solution possess certain properties 
peculiar to themselves which we class together as acid properties. 

iThus they neutralise bases, change the colour of certain indicators, are 
our to the taste, and so on. We are therefore disposed to attribute 
o them the possession of some common constituent which shall account 
)or these common properties. On asking what aqueous solutions of 
* he various acids have in common, we find for answer "hydrogen ions" 
f we adopt the hypothesis of electrolytic dissociation. Let us suppose 
he peculiar properties of acids to be due to hydrogen ions. How in 
hat case are we to explain the different strengths of the acids 1 
Evidently on the assumption that different acids in equivalent solution 
r ield different amounts of these ions. The acid which in a normal 
olution produces more hydrogen ions will be the more powerful acid, 
o that on this hypothesis the degree of dissociation of an acid 
urnishes a measure of its strength. But if we compare equivalent 
olutions of different acids under the same conditions, the electrical 
onductivity is closely proportional to the degree of dissociation of the 
lissolved substance. This arises from the fact that the speed of the 
tydrogen ion is much greater than the speed of any negative ion with 
rhich it may be associated. The conductivity, then, of any acid solu- 
tion is due principally to the hydrogen ions it contains, so that if we 
Compare the conductivities of solutions of different acids, the values 
ve obtain are nearly proportional to the relative amounts of hydrogen 
ons in the solutions, and thus to the relative strengths of the acids. 
>ince it is a very easy matter to measure the conductivity of solutions, 
his method of determining the relative strengths of acids has practi- 
cally superseded the other methods, especially in the case of the weaker 
>rganic acids. For them it is possible to calculate a dissociation con- 
tant according to the formula given on p. 224, and this constant is 
r ery generally accepted as a measure of their strength, for which 
eason it is sometimes spoken of as the affinity constant of the acids. 
It will be remembered that the theoretical dissociation formula only 
/pplies to half-electrolytes, the strong and highly dissociated mineral 
.cids giving only constants with the empirically modified formulae of 
foidolphi and van t' Hoff. These empirical constants might be used as 
* affinity constants " for the strong acids, since they, like the true disso- 
ciation constants, give a measure of the relative dissociations of different 
i,cids independent of the dilution, but as their significance is doubtful, 
md the degree of constancy attained is not after all very great for 
/he highly-dissociated acids, they have not so far come into general use. 



Since the degree of dissociation cannot go beyond 100 per cent, we 
have here a natural limit set to the strength of acids. The limit is 
reached at moderate dilutions for some of the monobasic acids, which 
are therefore the strongest acids which can be met with in aqueous 
solution. These are hydrochloric, hydrobromic, hydriodic, nitric, and 
chloric acids amongst the common inorganic acids; amongst the 
organic acids we have the alkyl sulphuric acids, such as hydrogen ethyl 
sulphate, and the sulphonic acids, such as benzene sulphonic acid 
or ethane sulphonic acid. No dibasic acid is as strong as these 
monobasic acids, sulphuric acid being the strongest of this type. 
The fatty acids, such as acetic acid, are very much weaker than 

It should be noted that at great dilutions the differences in 
strength between acids begin to disappear. At infinite dilution all 
acids are equally dissociated, so that no one will contribute more 
hydrogen ions than another, and consequently all acids under these 
conditions will have the same strength. As has already been indicated, 
no such state can actually be realised, but it is well to remember that, 
in general, differences in strength of acids are more marked in com- 
paratively strong solutions than in very dilute solutions. Thus we 
have the following numbers in the case of acetic acid and its chlorine 
substitution products, which represent the percentage degree of dis- 
sociation or the proportion of available hydrogen existing in the 
solution as ions, i.e. in the active state. 

Dilution =32 Litres 

128 Litres 

512 Litres 

Acetic 2 "4 



Monochloracetic 20 



Dichloracetic 70 



Trichloracetic 90 



At the dilution 32, trichloracetic acid has 37 times as many hydrogen 
ions as an equivalent solution of acetic acid ; at the dilution 512 it has 
only 11 times as many. It is true that between the dilutions 32 and 
512, acetic acid gains only 6*7 hydrogen ions, where trichloracetic acid 
gains 9, but this is on account of the great difference in strength 
between the acids, and it can be seen that between 128 litres and 
512 litres the actual gain of the acetic acid is greater than the gain 
of the trichloracetic acid, and this would be more and more evident 
as the dilution proceeded. In the case of the other acids which art 
more nearly equal in strength, the equalisation of strength as 
dilution proceeds is much more evident. From 32 litres to 512 litres 
trichloracetic acid only gains 9 hydrogen ions, where dichloracetic 
acid gains 28, and monochloracetic acid gains 37. At 32 litre? 
trichloracetic acid is nearly 30 per cent stronger than dichloraceti* 
acid ; at 512 litres their strengths are almost equal. 

It now remains to show the connection between the degree o 
dissociation of two acids and the proportion in which they will shar< 


a base between them, when both are competing for it in the same 
solution. Let the acids HA and HA", and the base NaOH, be dissolved 
in water so that 1 gram molecule of each is contained in v litres of 
the mixed solution. For the sake of simplifying the calculation, we 
shall also suppose that the acids are weak and obey Ostwald's dilution 
formula (p. 224), that their degree of dissociation at the dilution 
considered is very small, and that the dissociations of the sodium salts 
produced are equal, which will in fact be the case. Let x be the 
amount of the acid HA neutralised by the soda, then the quantity of 
HA' neutralised will be 1 -x, since the total quantity of soda is 1. 
If now h represents the quantity of hydrogen in the solution as ions, 
and d the common dissociation factor of the two sodium salts, we shall 
have the following quantities of the various substances existing in 
equilibrium with one another : — 

NaA' . . . x, of which (1 - d)x undissociated. 

NaA . . . (1 -x), of which (1 -d)(l -x) undissociated. 

HA . . . (1-aOi of which practically all undissociated. 

HA' . . . x, of which practically all undissociated. 

H ions . . h, a very small amount, from HA and HA'. 

Na ions . . d(x + l -x) = d, from NaA and NaA'. 

A ions . . . dx, from NaA almost entirely. 

A' ions . . d(l -x) t from NaA' almost entirely. 

Now in order that equilibrium may exist between the undissociated 
HA and the ions H and A, the requirements of Ostwald's dilution 
formula must be fulfilled, and we must have 

H ions x A ions 
(undiss. HA)v 

Substituting the values given in the above table, we obtain 


(1 - x)v 
Similarly for the acid HA' we get 

= k 

hd(\ - x) 


] Dividing the first of these equations by the second we have 

— = /*" 

l-x V k r 



{\-xf ¥ 

le. the ratio of the avidities of two acids is equal to the square root of 
the ratio of their dissociation constants, if all the conditions mentioned 
J above are fulfilled. 

We can get a direct relation between the avidities and the degree 



of dissociation at a given dilution by taking account of the dilution 
formulae for pure solutions of the acids. For the acids HA and HA' 
let the degrees of dissociation at the dilution v be m and m' respec- 
tively. We have then 

m 2 7 , m' 2 v 

r- r- = *> and,- -rr — k. 

(l-m)v (l-m)v 

Since the degree of dissociation of the acids is by hypothesis very 
small, 1 -m and 1 - m become very nearly equal to 1, and conse- 
quently by division we have approximately 

m 2 _k 
m 2 "' - J? ' 

But we found above 


x* k 

(l-xf k' 

x m 
— — _ , 

1 -x m 

i.e. ratio in which a base distributes itself under the conditions named 
between two acids, is practically equal to the ratio of the degrees of 
dissociation of the separate acids at the same concentration, or, in 
other words, is practically equal to the ratio of the electrical con- 
ductivities of the acids under similar conditions of dilution. The 
dissociation theory of Arrhenius, therefore, furnishes us with a 
satisfactory explanation of the connection between the various methods 
of measuring the strengths of acids ; the fundamental assumption 
being that the activity of acids, as acids, is due entirely to the 
presence of hydrogen ions, the various acids differing from each other 
only in the number of hydrogen ions produced when equivalent 
amounts are dissolved in a given quantity of water. The sour taste, 
the action on indicators, the accelerating effect on sugar inversion, etc., 
are all attributed to the hydrogen ions, and increase in magnitude as 
the number of hydrogen ions increases. 

The methods employed for measuring the relative strengths oi 
bases are in all respects similar to the methods adopted for acids. 
The distribution of an acid between two competing bases, the 
accelerating effect of different bases on a certain chemical action, and 
the electrical conductivities of equivalent solutions of the bases, have 
all been investigated, and have been found to lead to consistent 

If we inquire into what is common to solutions of all bases, w* 
find that the dissociation hypothesis gives us " hydroxy 1 ions" foi 
answer. Just as it attributed the essential properties of acids t( 
hydrogen ions, so it attributes the essential properties of alkalies t( 


hydroxyl ions. These are responsible for the alkaline taste, the action 
on indicators, and for the power of neutralising acids. Bases differ 
from one another in strength according as their equivalent solutions 
produce more or fewer hydroxyl ions, that base being the stronger 
which produces the greater number. 

The electrical conductivity of equivalent solutions of the soluble 
bases gives us a means of judging their relative strengths, but a 
reference to the table of ionic velocities on p. 212 will show that the 
value of the conductivity is not such a direct measure of the strength 
of bases as it is of acids, in view of the fact that the speed of the 
hydroxyl ion does not exceed the speeds of the positive ions in the 
same proportion as the speed of the hydrogen ion exceeds the speeds of 
the negative ions. The total conductivity of the base is therefore due 
to a less extent to the hydroxyl ion than the conductivity of an acid 
is due to the hydrogen ion. The degree of dissociation, however, if 
calculated from the conductivities in the usual way (p. 222), gives 
the correct measure of the strengths of bases when in equivalent 
solution, and the dissociation constant of bases has the same significance 
in this respect as the dissociation constants of acids, the stronger base 
being that with the greater constant. 

A direct velocity method, which has been applied to determining 
the strengths of bases, is the rate at which they saponify methyl 
acetate (cp. p. 256). The saponification is apparently effected by 
hydroxyl ions, and equivalent solutions of different bases will give 
velocity constants (at the beginning of the saponification at least) 
which will be proportional to the number of hydroxyl ions in the 
solution, i.e. to the strengths of the bases. The following numbers 
were obtained in fortieth normal solution of the various bases, the 
value of the constant for lithium hydroxide being made equal to 100 : — 

Lithium hydroxide 100 

Sodium hydroxide . . . . . . . 98 

Potassium hydroxide . 98 

Thallium hydroxide 89 

Tetraethylammonium hydroxide . . . . 75 

Triethylammonium hydroxide . . . . . 14 

Diethylammonium hydroxide 16 

Ethylammonium hydroxide 12 

Ammonium hydroxide ...... 2 

The strong alkalies, lithia, soda, potash, have practically reached the 
limit of strength, their dissociation at moderate dilutions being almost 
complete. The hydroxides of the metals of the alkaline earths are 
equally strong, as similar experiments have proved. Ammonia, or, 
as it exists in solution, ammonium hydroxide, is a comparatively feeble 
base, bearing much the same relation to the strong alkalies as the . 
weak organic acids to the strong mineral acids. The alkylammonium 
hydroxides are all stronger bases than ammonium hydroxide, the 
tetra-alkyl hydroxides being nearly comparable in point of strength to 


the caustic alkalies, as indeed is evident from their general chemical 

The only acceleration method hitherto found by means of which 
the relative strengths of bases may be determined is the transformation 
of the alkaloid hyoscyamine into the isomeric alkaloid atropine, the 
course of which can be followed with the polarimeter. The amount 
of the acceleration here seems to be proportional to the number of 
hydroxyl ions in the solution, but unfortunately the method cannot 
be applied with great strictness on account of secondary decompositions 
of the atropine, which interfere with the calculation of the velocity 
constant. The bases, potassium hydroxide, sodium hydroxide, and 
tetramethylammonium hydroxide, were found to have nearly equal 
accelerating effects, which is in agreement with the results of the 
saponification method. 

There are certain acids and bases so weak that it is difficult to 
apply some of the above methods for obtaining their relative strengths 
in aqueous solution. Their degree of dissociation at practically 
available dilutions is so slight that a direct measurement either of their 
accelerating action or their conductivity leads to doubtful results, 
whilst methods involving formation of their salts may also prove 
useless owing to the decomposing influence of water. Hitherto 
we have spoken of water as a perfectly neutral substance, but this is 
far from being the case. So long, indeed, as we deal with strong 
acids and bases, and the salts formed from them, the solvent water 
in which they are contained may be regarded as neutral, but if either 
very weak acids or very weak bases are in question, the chemical 
nature of the water must be taken into consideration. 

Ordinary tap- water has a very considerable electrical conductivity, i.e. 
it must contain ions in moderate quantity. The conductivity of distilled 
water is very much less, so we conclude that the conductivity of the 
tap-water is chiefly due to impurity. The more care that is devoted 
to the distillation of the water the less does its conductivity become, 
but it is difficult to procure and keep water with a conductivity at 
18° of less than 1 x 10 " 10 times that of mercury. Kohlrausch, 
however, on purifying water in platinum vessels by distillation in 
vacuo, and condensing the pure vapour directly in the resistance 
vessel, found that the purest water he could obtain had a conductivity 
of 0*036 x 10 " 10 . This conductivity must be accepted as the specific 
conductivity of pure water, for it is in close accordance with numbers 
calculated from the chemical and electrical behaviour of the substance. 
If the conductivity is not due to dissolved impurity, the ions with 
which the electricity travels must come from the water itself. Since 
water contains only hydrogen and oxygen, the ions we should expect 
are hydrogen ions and hydroxyl ions. This assumption accounts very 
well for the chemical and electrochemical behaviour of water, and 
affords the desired basis of a numerical comparison between the results 


of the different investigations. Corresponding to the very small con- 
ductivity, the degree of dissociation is very small. At 25° the amount 
of hydrogen contained in water in the form of ions is about 2 milli- 
grams per ton, the amount of hydroxyl being 1 7 times as great as this. 
These amounts, although excessively minute, are sufficient to confer 
on the water the properties of a weak acid on account of the hydrogen 
ions, and of a weak base on account of the hydroxyl ions. Chemically 
speaking, these properties are most evident in the phenomena of salt 
hydrolysis. If a salt, when dissolved in water, is only affected by the 
solvent in so far as electrolytic dissociation is concerned, its solution 
is neutral, since there are in it neither free hydrogen ions nor free 
hydroxyl ions in appreciable quantity. This we find to be the case 
with salts derived from strong acids and strong bases, e.g. sodium 
chloride, potassium sulphate, and the like. If, on the other hand, 
we dissolve in water a salt formed by the neutralisation of a strong 
acid by a weak base, the solution has a distinctly acid reaction, i.e. 
must contain free hydrogen ions in quantity. Examples of such salts 
are the chlorides or nitrates of aluminium, copper, and zinc among 
inorganic compounds, and the same salts of aniline, pyridine, and 
urea amongst organic compounds. If the base is as weak as 
diphenylamine, treatment with water is sufficient to decompose it 
almost entirely into free acid and free base (cp. p. 247). 

If the salt is formed from a strong base and a weak acid, it is 
equally decomposed by water, but the reaction of the solution is then 
alkaline, as in it there is an excess of hydroxyl ions. Examples of 
such salts are to be found in sodium or potassium borate, carbonate, 
or cyanide, and in the soaps. 

In order to understand how salts of the above types come to 
possess an acid or an alkaline reaction, we have to consider the relative 
degrees of dissociation of the acid and base liberated from the salts by 
the action of the water. Suppose the salt to be formed by the neutral- 
isation of a strong base, say caustic soda, by a weak acid, say carbonic 
acid. We may assume provisionally that the salt is decomposed by 
the water according to the following equation : — 

Na 2 C0 3 + 2H 2 = 2NaOH + H 2 C0 3 . 

The base and acid are produced by the decomposition in equivalent 
quantities, but the base, being a strong one, is highly dissociated, 
whilst the acid, being a weak one, is scarcely dissociated at all. From 
the base, then, are produced hydroxyl ions in comparatively large 
quantity, while the acid supplies very few hydrogen ions. There is 
thus a great excess of hydroxyl over hydrogen ions, and the solution 
has in consequence an alkaline reaction. This is all the more marked 
inasmuch as the hydrolysis takes place, in all probability, according to 
the equation 

Na 2 C0 3 + H 2 = NaHC0 3 + NaOH, 


the acid salt being formed instead of the free acid. Now the acid 
salt supplies no hydrogen ions at all, as far as we can judge, so that 
practically the whole of the hydroxyl ions produced by the hydrolysis 
remain in the solution as such, and thus impart to it a strong alkaline 

With such a salt as aniline hydrochloride the reverse is the case. 
The free base, phenylammonium hydroxide, produced according to 
the equation 

NH 3 (C 6 H 5 )C1 + H 2 = NH 3 (C 6 H 5 )OH + HC1, 

is very feeble, and consequently yields few hydroxyl ions. The acid 
produced, on the other hand, is almost fully dissociated, so that the 
solution will have a very large excess of hydrogen ions, and a corre- 
spondingly strong acid reaction. 

If we take a series of weak bases combined with the same acid, 
say hydrochloric acid, to form salts, and measure the amount of 
hydrolysis in equivalent solutions of these salts, we can form an 
estimate of the relative strengths of the bases. The weaker the base 
is, the more of it will be expelled from combination with the acid by 
the competing base water. The greater, then, the amount of hydrolysis 
in a series of salts under the same conditions, the weaker is the base 
which is in combination with the acid. 

The determination of the extent of the hydrolysis is often some- 
what difficult. It might be thought that the free acid could be 
titrated with an alkali and an indicator, but a little consideration 
shows that this is impossible. Suppose that the salt were hydrolysed 
to the extent of 5 per cent. As soon as we have neutralised that 
5 per cent of acid by a strong base, the originally undecomposed pro- 
portion of the salt is hydrolysed by the action of the water, and more 
acid is produced. Although this in turn is neutralised by the addition 
of a strong base, the process of hydrolysis still goes on, and the solu- 
tion will not become neutral until all the acid which was originally 
combined with the weak base has passed into combination with the 
strong base with which we titrate the solution. A solution of aniline 
hydrochloride behaves towards caustic alkali and phenol -phthaleine 
exactly like an equivalent solution of hydrochloric acid, owing to the 
progressive hydrolysis of the salt. 

In determining the extent of hydrolysis, then, a method must be 
employed which will not disturb the hydrolysis equilibrium. Measure- 
ments of optical or other physical properties have been employed with 
some degree of success, and also reaction velocity methods. For 
example, we can tell approximately how much free hydrochloric acid 
there is in a solution of aniline hydrochloride or urea hydrochloride, 
by ascertaining at what rate they catalyse methyl acetate or invert 
cane sugar (cp. p. 271). The velocity constant for the catalysis of 
methyl acetate is approximately proportional to the amount of free 


hydrochloric acid in the solution, so that a determination of the one 
leads to a knowledge of the other. Similarly, we can ascertain how 
much free caustic soda there is in a solution of sodium carbonate by 
finding at what rate the solution saponifies ethyl acetate, the initial 
rate of saponification being nearly proportional to the amount of free 
alkali in the solution. 

The following tables will serve to indicate the extent of hydrolysis 
of some common salts of weak bases and weak acids : — 

Percentage Hydrolysis of Hydrochlorides of Weak Bases 
jE = 25°; v = 32. 

Aniline 2*6 

Paratoluidine . . . . . . 1*5 

Orthotoluidine 3*1 

Urea 76 

The urea hydrochloride undecomposed by the water apparently only 
amounts to one-fourth of the whole. The hydrolysis of the salts of 
the aromatic bases, on the other hand, is comparatively slight, and 
experiments on the rate of sugar inversion at 80° indicate that the 
hydrolysis suffered by the hydrochloride of a weak inorganic base 
like alumina is of the same order of magnitude. 

Percentage Hydrolysis of Salts of "Weak Acids 
t=25°; *?=10. 

Potassium phenolate 3*1 

Potassium cyanide 1*1 

Borax 0*5 

Sodium acetate *01 

Acetic acid is usually spoken of as a weak acid, but the above table 
shows that it is much more powerful than any of the other acids 
mentioned, the hydrolysed part only amounting to a ten-thousandth 
of the whole under the specified conditions. 

The extent of hydrolysis of a salt increases with increasing dilu- 
tion, as is evident from a consideration of the equilibrium. According 
to the law of mass action, 

k x act. mass of salt x act. mass of water = k' x act. mass of acid x 
act. mass of base. 

Now for dilute solutions the active mass of the water is a constant, 

so that 

act. mass of acid x act. mass of base 

— - = constant. 

act. mass ot salt 

Let the total amount of material considered be 1 gram molecule dis- 
solved in v litres of water, and let the hydrolysed portion be x. Then 


fx\ 2 1 - x a" 

vJ ' v (1 -x)v 

= constant. 

As v increases, x must, according to the formula, increase likewise, 
and if the extent of hydrolysis x is small, it will increase very nearly 
proportionally to the square root of the dilution, as may be shown in 
the manner adopted for the closely similar dissociation equilibrium 
(p. 225). 



In the chapter on balanced action we have seen that when the active 
mass of one or more of the products of a dissociation is increased, 
the degree of dissociation is diminished (p. 250). This rule is especi- 
ally important when we deal with solutions of salts, acids, and bases, 
all of which are electrolytically dissociated, and that to very different 
degrees. In cases of gaseous dissociation we can usually add one of 
the products of dissociation without adding anything else at the same 
time. This cannot be done with dissolved electrolytes, for the nature 
of the dissociation is such that the solution must always remain 
electrically neutral, although the products of dissociation are electric- 
ally charged. When, therefore, we add one of the products of dis- 
sociation to an electrolytically dissociated substance, we are compelled 
to add at the same time an electrically equivalent quantity of an ion 
oppositely charged. Thus if we consider a solution of hydrogen 
acetate, we find that we can only add hydrogen ions by adding an 
acid, say hydrochloric acid, which not only contributes hydrogen ions, 
but choride ions as well. Similarly, if we wish to increase the amount 
of acetate ions in the given volume, we can only do so by adding to 
the solution an acetate, which yields metallic ions at the same time as 
it yields acetate ions. Notwithstanding this complication, however, 
the equilibrium equation for hydrogen acetate still remains the same, 
namely — 

act. mass H* x act. mass C 2 H 3 2 ' = const, x act. mass undiss. C 2 H 4 2 , 

whether a small quantity of another ion is added or not, so that if we 
increase the active mass of either the hydrogen ion or the acetion, the 
active mass of the undissociated hydrogen acetate is also increased, 
i.e. the degree of dissociation is diminished. 

As has already been indicated, the effect on the degree of dis- 
sociation is greater when the substance considered is only slightly 
dissociated. Now acetic acid, even in moderately dilute solution, is 
only feebly dissociated, so that the addition of an equivalent quantity 


of a strongly dissociated acid like hydrochloric acid practically reduces 
the dissociation of the acetic acid to zero. Suppose, for example, that 
at the given dilution hydrochloric acid is fifty times more dissociated 
than acetic acid, the addition of an equivalent of the former will 
practically reduce the quantity of acetions to a fiftieth part of the 
original value, for it has multiplied the number of hydrogen ions 
about fiftyfold, and the product of the active masses of the ions must 
remain practically constant, the active mass of the undissociated hydro- 
gen acetate suffering but little change from the diminution of the 
dissociation. The degree of dissociation is thus reduced to about a 
fiftieth part of its former magnitude. A similar reduction takes 
place if we add an equivalent quantity of an alkaline acetate to a 
solution of acetic acid. Although the acid itself is feebly dissociated, 
its salts are as highly dissociated as those of strong acids, with the 
result that the number of the acetions is greatly increased, and the 
number of the hydrogen ions correspondingly diminished. This 
reduction of the degree of dissociation of weak acids by the addition 
of their neutral salts to the solution is of great practical importance, 
as the number of their hydrogen ions, and consequently their activity as 
acids, is thereby greatly reduced (Chap. XXI V.). 

The addition to a strong acid of an equivalent or greater quantity 
of one of its neutral salts has very little effect on its activity as an 
acid, as measured by the proportion of hydrogen ions to which it 
gives rise. This is due to the acid as well as the salt being almost 
completely dissociated. If, for example, we add an equivalent of 
sodium chloride to a solution of hydrogen chloride, we at most 
double the number of chloride ions. The change on the undissociated 
amount is, however, also relatively great, so that to fulfil the require- 
ments of the equilibrium formula a very small diminution of the 
number of hydrogen ions is necessary, and consequently the activity 
of the acid is little affected. 

What holds good for acids likewise holds good for bases, the strength 
of which is measured by the proportion of hydroxyl ions derived from 
them. If to a feebly dissociated solution of ammonium hydroxide we 
add ammonium chloride, which is highly dissociated, we greatly augment 
the number of ammonium ions, and diminish to a corresponding extent 
the number of hydroxyl ions necessary for equilibrium. The base 
ammonium hydroxide, then, loses much of its activity when accom- 
panied in solution by ammonium salts. Strong bases like potassium 
hydroxide are only slightly affected by the addition of a neutral salt 
yielding the same metallic ion, for although the relative change on the 
undissociated proportion may be great, the same actual change has a 
very slight effect on the dissociated proportion, and thus leads to only 
a slight diminution in the number of hydroxyl ions. 

The addition of neutral salts to a dibasic acid like sulphuric acid 
is a much more complicated phenomenon than the addition of neutral 


salts to monobasic acids. This is chiefly due to the formation of acid 
salts, which dissociate rather as salts than as acids. Thus when we 
add an equivalent of sodium sulphate to a solution of hydrogen 
sulphate, a certain proportion of the two original salts remains in the 
solution, accompanied however by the intermediate acid salt, sodium 
hydrogen sulphate, NaHS0 4 . The sodium sulphate dissociates largely 
into the ions Na* and SO" ; the sulphuric acid dissociates into H* and 
SO/, but also produces ions HSO/ ; the sodium hydrogen sulphate, 
finally, dissociates chiefly into Na* and HSO/. There are therefore 
three undissociated substances in solution, viz. H 2 S0 4 , Na 2 S0 4 , and 
NaHS0 4 ; and at least four kinds of ions, viz. Na, H', HSO/, and SO/, 
so that the equilibrium is somewhat complex. 

In this connection it may be stated that dibasic acids very generally 
dissociate in solution into one hydrogen ion and the residue of the 
molecule, the second replaceable hydrogen atom not splitting off as an 
ion until the greater quantity of the first has been removed. Thus 
sulphuric acid in fairly strong solution in all probability contains very 
little of the ion SO/, the dissociation being principally into H' and 
HSO/ At greater dilutions, however, the ion SO/ appears in 
quantity, probably owing to the splitting up of the ion HSO/ into H* 
and SO/. One result of this is that weak dibasic acids give a dis- 
sociation constant of exactly the same character as that of a monobasic 
acid, the formula used in deriving which assumes that the dissociation 
takes place into two ions only. Thus malonic acid does not primarily 
dissociate according to the equation 

rn /COOH_ p /COO' H . 
U1±2 \COOH ~ un 2\C00' + " U ' 

but according to the equation 

rR /COOH_ pH /COO' +n . 
° 2 \COOH " un2 \COOH + • 

The equilibrium is therefore of exactly the same type as the dissocia- 
tion equilibrium of acetic acid, and follows the same law. 

This is seen in the following table, which gives the dissociation 
constant for malonic acid (cp. p. 224) : — 





























ncv of t\\ 

le expression 


iooz-= 100m2 i 

n the 1 

( 1 - m)v 


shows that the primary dissociation is into two ions and not into 

three, in which case the expression 7- r- 9 would be constant. It 

' (1 — m)v A 

will be noticed that after 50 per cent of the first hydrogen has split 

off as ion, the constant begins slightly to rise. The rise probably 

indicates that the second hydrogen atom is now being affected, i.e. 

that the action 

pn /COO' _ 0H /COO' 

on 2\COOH ~ u ^ 2 \COO' + ri 

has commenced to be appreciable. The exact point at which this 
second action begins varies very much with different acids. Many 
acids show the secondary dissociation when the primary has proceeded 
to the extent of about 50 per cent; but some acids, male'ic acid for 
example, give a good constant up to 90 per cent primary dissociation. 
For acid salts one may say that almost invariably the primary 
dissociation is into metallic ion and the rest of the molecule, no 
hydrogen ions appearing until this primary dissociation is far advanced, 
although, as indicated above, hydrogen ions may arise from the acid 
itself produced by the action of the water. Thus sodium hydrogen 
malonate dissociates primarily according to the equation 

pH /COONa_ rn /COO' ~. 
O±l2 \C00H ~ on2 \COOH + iNa > 

and the acid character of the solution, judged by the methods of the 
preceding chapter, is very feebly marked. 

In general, when we mix two electrolytic solutions, we cannot 
calculate the conductivity of the mixed solution from those of the 
components by the simple alligation formula, 1 because each dissolved 
substance affects the dissociation of the other, and thus alters the 
number of carrier ions. From the law of mass action, however, as 
applied to electrolytic equilibrium, we ascertain that there must be 
certain solutions which can be mixed together without alteration in 
the number or nature of the ions, and therefore without change in the 
average conducting power. Such solutions are called isohydric, and 
we shall first investigate the conditions for isohydry in the case of 
two electrolytes giving rise to a common ion, say HA and HA', each 
of which obeys Ostwald's dilution law. Let the dilutions of the two 
isohydric solutions be v and v respectively, and their degrees of dis- 
sociation m and m. For the acid HA we have the equilibrium 

1 If two substances when mixed retain their specific values of a property unchanged 
by the process of mixture, the specific value of the same property for the mixture can be 

calculated by the alligation formula =— , in which a and b represent the proportions 

of the components, and A and B the specific values of the property for the components. 


m 2 _ , 
(l-m)v~ ' 

and for the acid HA' the corresponding equation 

(1 - my 

= h\ 

If now we mix these isohydric solutions, the volume becomes v + v, 
and the number of hydrogen ions m + m\ For the acid HA under the 
new conditions we have now the equilibrium equation 

(m + m)m _ 
(1 - m)(v + v') " 

Dividing this equation by the first, we obtain 

(m + m')v 

1, or 

m + m' 


+ v' 

(y + v')m 




m " 


— 9 




v " 



Now — is the concentration of hydrogen ions in the acid HA, and — 

is the concentration of hydrogen ions in the isohydric solution of the 
acid HA', and these two concentrations prove to be equal. We may say, 
therefore, that solutions of electrolytes containing a common ion are 
isohydric when the concentration of the common ion in the different 
solutions is the same. 

It is often convenient for purposes of calculation to imagine an 
actual mixed solution to be split up into its component isohydric 
solutions, which may then be ideally mixed at any time without any 
change in the dissociation of the electrolytes occurring. For example, 
a solution containing equivalent quantities of hydrogen acetate and 
sodium acetate may be imagined to exist in a rectangular vessel with 
a movable vertical partition through which water can freely pass, but 
not the dissolved substances. Let the hydrogen acetate be on one 
side of the partition and the sodium acetate on the other. The 
partition is now to be moved until the concentration of the common 
ion, acetion, is the same on both sides. Since the sodium acetate is 
highly dissociated, it must receive most of the water in order that the 
concentration of the acetion may be as small as that derived from the 
slightly dissociated hydrogen acetate. The position of the partition 
for isohydry must therefore be as shown in Fig. 42, which represents 
a horizontal section of the vessel, the liquid rising to the same level 
on both sides of the diaphragm. It must be borne in mind that con- 


centrating the solution of hydrogen acetate does not involve similar 
concentration of the hydrogen or acetate ions, for as the dilution 

diminishes, the degree of dissocia- 
tion, and therefore the proportion 
of ions, diminishes also, although 
at a smaller rate. Beginners are 
apt to reason that if the solution 
of one electrolyte is ten times 
more dissociated than an equival- 
Fia 42 - ent solution of another, it is only 

necessary to concentrate the second solution to a tenth of its volume 
in order that the degrees of dissociation of the two solutions may 
become equal. In view of the diminution of the degree of dissocia- 
tion as the dilution diminishes, a much greater degree of concentra- 
tion is necessary. 

If we consider the mixing of two salts which have no common ion, 
say NaCI and KBr, the problem becomes much more complicated. 
When the two salts are mixed, we have not only the substances origin- 
ally in solution, i.e. the undissociated salts NaCI and KBr, and their 
ions, Na, K, CI, and Br, but also the new undissociated substances 
NaBr and KCI. Let there be prepared isohydric solutions of the 
different salts, NaCI being made isohydric with NaBr, by getting the 
sodium ions of the same concentration in both solutions ; KCI may 
then be made isohydric with NaCI by making the chloride ions of the 
same concentrations in the two solutions. KBr may finally be made 
isohydric with KCI by equalising the concentration of the K ions. 
Any two of these solutions then which possess a common ion may 
be mixed in any proportions without change in the dissociation. If we 
wish to mix all four, we must take 
volumes of the solutions such that 
the products of the volumes of 
reciprocal pairs are equal. If a, b, 
c, d be the volumes of the iso- 
hydric solutions of NaCI, NaBr, 
KCI, and KBr respectively, such 

ad = be, 

then the solutions may be mixed 
in these proportions without change 
in the dissociation. Using a dia- 
grammatic representation similar 
to that adopted for mixtures of 
pairs of electrolytes with a common 

ion, we get the diagram Fig. 43, which fulfills the desired condition. 
The proof of the condition may be given on the supposition that 






the substances obey Ostwald's dilution law. Let a, b, c, d be the 
volumes of the isohydric solutions which when mixed will produce no 
change in the dissociation. Before mixing, the equilibrium of the 
sodium chloride will, assuming its quantity to be unity, be represented 
by the formula 

m m 

a a m 2 _, 

1 - m (1 -m)a 

where m is the degree of dissociation. Let the solutions be now mixed. 

The quantity of the sodium ion has now increased in the ratio of a + b to 

a, since b volumes of NaBr have been added to the original a volumes of 

NaCl, and the concentration of the sodium ions is the same in both 

solutions. But the volume in which this quantity is contained is 

now a + b + c + d, so that the active mass of the sodium ion is now 

a + b 1 m(a + b) . ., . . . , 

m x x r =~t t — ^- Similarly the quantity of 

a a + b + c + d a(a + b + c + d) J ^ J 

the chloride ion increases in the ratio of a + c to a, and its active mass 

becomes — \ -. The undissociated proportion of sodium 

a(a + b + c + d) r r 

chloride remains the same as before, viz. 1 - m. We have therefore 

for the new equilibrium the equation 

m(a + b) m(a + c) 

a(a + b + c + d)' a(a + b + c + d) 

1 -m 



whence, since k is also equal to 7- r- ? 

^ (1 -m)a 

m 2 (a + b)(a + c) m 2 

( 1 - m)d\a + b + c + d) (1- m)a 
{a + b)(a + c) 
a(a + b + c + d) ~~ ' 
a 2 + ab + ac + be = a 2 + ab + ac + ad, 
be = ad, 
which was to be proved. 

According to Guldberg and Waage's Law, we should have for 
equilibrium in the balanced action 

NaCl + KBr = NaBr + KCl 

^the expression 

[NaCl] x [KBr] 

r XT p i — hF™4 = constant, 

[NaBr] x [KC1] 



where the formulae in square brackets represent the active masses of 
the respective substances. This equilibrium formula takes no account 
of electrolytic dissociation of the various salts, and is only valid under 
certain conditions of dissociation. The correct formula is 

[ diss. NaCl] x [diss. KBr] 
[diss. NaBr] x [diss. KC1] ' 

as may be deduced from the above relation ad = be. Since all the 
solutions to which these letters refer are isohydric in pairs, i.e. have 
the same concentration of ions, the volumes a, b, c, d are proportional 

to the quantities of the ions 
in the various solutions, i.e. 
to the dissociated quantities 
of the salts, and not to their 
total quantities. When all 
the substances are highly 
dissociated, Guldberg and 
Waage's Law leads to very 
nearly the same result as 
when the dissociation is con- 
sidered, and the same holds 
true when two of the four 
substances are highly disso- 
ciated. When, however, one 
or three of the substances 
are highly dissociated, there 
is usually a great discrepancy 
between the two modes of 
calculating the equilibrium, 
longer even approximately 










Fig. 44. 


Guldberg and Waage's Law being 
true, except in special circumstances. 

As an example of the application of the theory of isohydric solu- 
tions as applied to the equilibrium of four dissociated substances, we 
may take the distribution of a base between two acids obeying 
Ostwald's dilution law and find the relation of the distribution 
ratio to the ratio of the dissociation constants of the acids. Let, as 
before (p. 275), one gram molecular weight of each of the substances 
HA, HA', and NaOH be dissolved in a certain volume of water, and 
let the solution thus obtained be ideally split up into isohydric 
solutions of the same ionic concentration i. We thus get the diagram 
Fig. 44. 

The volumes are again represented by a, &, c, d, and the following 
table gives the data necessary for the calculation, if x is the amount of 
HA neutralised by the soda : — 






























Total quantity 
Dissociated quantity 

Degree of dissociation (m) 
Dilution (v) 

is the concentration of the ions is the same in all the solutions, the 
Associated quantities are equal to the volumes of the solutions multi- 
plied by the common ionic concentration, L The degree of dissociation, 
n, is the ratio of the dissociated to the total quantity, and the volume 
livided by the quantity contained in it gives the volume which contains 
mit quantity measured in gram molecules, i.e. the dilution v. The 
iicids by supposition obey the theoretical dilution law 

(1 - m)v 

By a simplification which has already been adopted when the degree 
of dissociation is small, we may neglect m in comparison with 1, and 

write the dilution formula — = k Now, substituting the above values 

of m and v for the acid HA, we obtain 





a 1-x 

ad similarly for the acid HA', we obtain 

i 2 c 


ivision then gives 

c(l-x) k' 
Now for equilibrium we have 

ad = be, or ajc = b/d 9 
bx k 

Whence -rrz r = T7* 

d(l -x) k 
As before, we may assume that the two sodium salts are dissociated 


to an equal extent, so that in their case the ratio of the dissociated 
quantities is the ratio of the total quantities, i.e. 

ib _ x 
id 1 -x 

We thus obtain finally the relation 


x 2 _k X /Jc 

- xf " £" 0r 1 - x " V Jc' ' 

that is, the ratio of distribution of the base between the two acids is 
equal to the ratio of the square roots of the dissociation constants of 
the acids, a result already obtained on p. 275 under the same assump- 

The student who desires to familiarise himself with the equilibrium 
electrolytes in solution is advised to study the subject from the point 
of view of isohydric solutions, in particular when dealing with two 
electrolytes containing a common ion, or with double decompositions 
between electrolytes. In this last case the important fact to bear in 
mind is that the product of the dissociated quantities on one side of 
the equation is equal to the product of the dissociated quantities on 
the other. If, for example, we are dealing with the double decom- 

CH 3 . COONa + HC1 ^ CH 3 . COOH + NaCl, 

and the quantities of these substances, when equilibrium has been 
attained, are m v ra 2 , ra 3 , m 4 , with the degrees of dissociation d v d 2 , d s , d 4 
in the mixed solution, we have always the relation 

m^ x m 2 d 2 = m z d z x m^d^ . 

This relation we can combine with our knowledge of the general 
nature of the dissociation of the various substances, and its variation 
with dilution, to ascertain the actual character of the equilibrium. 
Thus Arrhenius, to whom the theory is due, has shown that the 
avidities of two monobasic acids at a given dilution are approximately 
proportional to the degrees of dissociation which they would have if 
each were dissolved separately in the given volume of solvent. This 
we showed above to be the case for two weak acids, but it is equally 
true if both acids are strong, or if one is strong and the other weak. 
In the case of dibasic acids, like sulphuric acid, the theory cannot 
easily be applied owing to the excessively complicated nature of the 
equilibrium caused by the presence of acid salts (cp. p. 285). 

It is easy, too, to prove from the theory that the degree of 
dissociation of a weak acid in presence of one of its salts is nearly 


nversely proportional to the quantity of salt present. If the weak 
1 icid should be in presence of several strongly dissociated electrolytes, 
t can also be shown that its degree of dissociation will be the same as 
f the dissociated parts of these electrolytes were the dissociated parts 
)f a salt of the acid itself. 

We have now to examine the nature of the equilibrium between 
;,he aqueous solution of a salt and the solid salt itself. To each 
emperature there corresponds a certain solubility of the salt, i.e. a 
ertain osmotic pressure of the dissolved substance in the solution 
rhich is in equilibrium with the solid. Now this osmotic pressure is 
aade up of more than one component : it is the sum of the partial 
•ressures of the undissociated salt and of the ions derived from the 
alt. The question thus arises: Is it the total osmotic pressure in 
he solution which directly regulates the equilibrium with the solid 
alt, or the osmotic pressure of the undissociated dissolved salt, or 
nally the osmotic pressure of the ions *? The most probable reply to 
i his question is that it is the osmotic pressure of the undissociated 
* ubstance which directly determines the equilibrium, and there are 
aany facts which support this conclusion. The undissociated salt 
.ere plays the part of intermediary between the ions and the solid : 
; is in equilibrium with the ions on the one hand and with the un- 
associated solid on the other. Considered in this aspect, the constant 
otal solubility of a solid salt at a given temperature in water is due 
3 the constant concentration of undissociated substance in the solution, 
diich in its turn is in equilibrium with a constant concentration of 
! he ions. It is possible, however, to supply additional quantities of 
-ne or other of these ions to the solution, and we have to inquire 

Iljfcito the effect this will have on the solubility equilibrium. 
In order to secure conditions favourable for calculation and for 
sperimental verification of the results deduced from the theory, it is 
dvisable to consider the equilibrium in the case of a sparingly soluble 
[alt, so that the solutions considered are dilute. As an example we 
lay take silver bromate, AgBr0 3 , the concentration of the saturated 
tiution of which at 24*5° is 0*0081 normal. The primary equilibrium 
hich determines the solubility is here supposed to be that between 
le solid silver bromate and the undissociated silver bromate in the 
)lution. The concentration of this last will remain constant if the 
imperature remains at 24*5° and the solvent remains water. The 
idition of a small quantity of a perfectly neutral substance, such as 

Icohol, sugar, and the like, does not appreciably affect the solubility 
i any substance in water, since the nature of the solvent practically 
jmains the same. Besides the undissociated silver bromate in the 
>lution, we have silver ions and bromate ions. We can increase the 
I mcentration of silver ions by adding a soluble silver salt, and we 
in increase the concentration of bromate ions by adding a soluble 
Tomate. Suppose that we add such a quantity of silver nitrate as to 


double the number of silver ions after equilibrium has been attained 
The concentration of the undissociated silver bromate will, by hypothesis 
remain the same as before. But the dissociation equilibrium of silvei 
bromate requires that the product of the concentrations of the ion? 
will be equal to a constant into the concentration of the undissociatec 
salt, i.e. will remain constant. If, therefore, the concentration of th( 
silver ions is doubled, the concentration of the bromate ions must b< 
halved, in order that the product of the two may have the same valu< 
as before. Bromate ions can only fall out of solution along with ar 
equivalent quantity of some positive ion, and since the only kind o 
positive ion in the solution is the silver ion, bromate of silver must b* 
precipitated in order to re-establish equilibrium. The effect, then, o 
adding silver nitrate to the silver bromate solution is to diminish th* 
solubility of the silver bromate, and that in a degree depending 01 
the amount of silver nitrate added. The addition of a soluble bromati 
acts in precisely the same way. The number of bromate ions a 
equilibrium is increased, and the number of silver ions must b< 
proportionately diminished in order to secure the constancy of th< 
product of the bromate and silver ions. 

The following numerical example will afford an insight into th< 
mode of calculation. As has already been stated, the concentration o 
a saturated silver bromate solution at 24*5° is 0*0081 normal. If w» 
assume the salt to be entirely dissociated, the product of the ions is 

0-0081 x 0-0081 = 0*0000656. 

Now a quantity of silver nitrate is added, which, when dissolved h 
the same water as contains the silver bromate would make the solutioi 
0*0085 normal with respect to silver nitrate. Again we assume tha 
the silver nitrate is entirely dissociated. Suppose that the concentra 
tion of the silver bromate now remaining in the solution is x, a smalle 
quantity than before. The concentration of silver ions will then h 
0*0085 + x, and the concentration of bromate ions will be x. We hav< 
therefore the product of these concentrations equal to the former pro 
duct, i.e. 

(0-0085 + x)x = 0-0000656, 

x = 0*0049. 

We should consequently expect the addition of silver nitrate to reduc< 
the strength of the saturated solution of silver bromate from 0*008! 
to 0*0049. An actual determination showed that the solubility wa: ' 
reduced to 0*0051, which is in fair agreement with the theoretica 
result. It must be noted, however, that the theoretical result wa; 
deduced on the erroneous assumption that the degree of dissociatioi 
of the various substances remained the same throughout the experi 
ments. This is, of course, not the case, as the degree of dissociatioi 


at the dilutions considered is not equal to unity, and is diminished on 
the addition of the silver nitrate. It is easy, however, to take account 
of the change in the degree of dissociation of the silver salts by making 
use of conductivity determinations, although the formula for equilibrium 
then becomes somewhat complicated. Making the necessary corrections 
in the above case, the theoretical number comes out equal to 0*00506, 
which is very nearly the value observed for the solubility. Since 
silver nitrate and sodium bromate have practically the same effect so 
far as dissociation is concerned, we should expect equivalent quantities 
of these two salts to diminish the solubility of silver bromate equally. 
Experiment shows that an amount of sodium bromate equivalent to 
the silver nitrate added in the above experiment diminishes the 
solubility to 0*0052, a value very nearly identical with the former 

When two sparingly soluble salts yielding a common ion are 
shaken up with the same quantity of water, each diminishes the 
solubility of the other in a degree which can be calculated as in the 
previous instance. Thus the saturated solutions of thallium chloride, 
TIG, and thallium thiocyanate, T1SCN, have a concentration of 
0*0161 and 0*0149 respectively in gram molecules per litre. For the 
constant product of ionic concentrations we have, therefore, 0*01 6 1 2 and 
0*0149 2 , if each is fully dissociated into ions. If the solubility of the 
chloride in presence of the thiocyanate is x, and the solubility of the 
thiocyanate in presence of the chloride is y, these two numbers give 
the concentrations of the chloride and thiocyanate ions respectively, 
while their sum, x + y, gives the concentration of the thallium ion. We 
thus obtain the simultaneous equations 

x(x + y) = 0'0Ul\ 
y(x + y) = 0-QU9 2 , 

whence x — 0*01 18, and y= 0*0101, the numbers found by experiment 
being in good agreement, viz. 0*0119 and 0*0107. By taking account 
of the actual degree of dissociation, the harmony between the experi- 
mental and calculated values is even more marked. The lowering of 
the solubility of an electrolyte by the introduction into the solution 
of another electrolyte possessing a common ion with the first is a 
phenomenon of very general occurrence, the only exceptions being 
when the two substances form a double salt or act on each other 
chemically in the solution. Instances of the application of the 
theoretical results will be given in the next chapter. 



When many of the ordinary chemical reactions are looked upon from 
the standpoint of the theory of electrolytic dissociation, they present 
an aspect very different from that to which we are accustomed. The 
neutralisation of strong acids by strong bases in dilute solution is a 
typical example. If the acid is hydrochloric acid, and the base sodium 
hydroxide, we have the equation 

HC1 + NaOH = NaCl + HOH. 

Now on the dissociation theory all the substances concerned in this 
action are highly dissociated in aqueous solution, the water itself being 
the only exception. Writing, then, the equation for the ions, we 

H' + CI' + Na + OH' = Na + CI' + HOH, 

or, eliminating what is common to both members of the equation, 

H" + OH' = HOH. 

The neutralisation of a strong acid by a strong base consists then 
essentially in the union of hydrogen and hydroxyl ions to form water. 
So long as the base, acid, and salt are fully dissociated, their nature 
makes no difference whatever on the character of the chemical act of 
neutralisation. This we find to be in conformity with many 
experimental facts. For example, the heat of neutralisation of one 
equivalent of a strong acid in dilute solution by a corresponding 
quantity of a strong base is very nearly 137 K, as the following table 
shows, the base used being caustic soda : — 

Acid. Heat of Neutralisation. 

Hydrochloric 137 K 

Hydrobromic 137 ,, 

Hydriodic 137 „ 

Chloric 138 ,, 

Bromic 138 ,, 

Iodic 138 ,, 

Nitric 137 ,, 


A similar table for the heats of neutralisation of bases by an 
equivalent of hydrochloric acid shows that the heat of neutralisation 
is independent of the base, as long as it is fully dissociated. 

Lithium hydroxide 
Sodium hydroxide 
Potassium hydroxide 
Thallium hydroxide 
Barium hydroxide . 
Strontium hydroxide 
Calcium hydroxide 
Tetramethylammonium hydroxide 

Heat of Neutralisation. 
138 K 

137 „ 

138 „ 

139 „ 

138 „ 

139 „ 
137 „ 

When we deal with weak acids or weak bases the heats of neutralisa- 
tion often diverge greatly from the mean value for the strongly 
dissociated substances. Thus the heats of neutralisation of some 
comparatively feebly dissociated acids by sodium hydroxide are given 
in the following table, the acids not being so feeble, however, as to 
suffer sensible hydrolysis in aqueous solution (cp. p. 278) : — 


Metaphosphoric acid 
Hypophosphorus . 
Hydrofluoric acid . 
Acetic acid 
Monochloracetic acid 
Dichloracetic acid . 

Heat of Neutralisation. 
143 K 
151 „ 
134 „ 
143 „ 
148 ., 

Corresponding numbers for weak bases neutralised by hydrochloric 

acid are : 

Ammonium hydroxide . 
Methylammonium hydroxide . 
Dimethylammonium hydroxide 
Trimethylammonium hydroxide 

Heat of Neutralisation. 
122 K 
. 131 „ 
118 „ 

87 „ 

The explanation of these divergences from the value for highly dis- 
sociated substances is simple. Ammonium hydroxide is only feebly 
dissociated at the dilution considered ; the chemical action is not 

H' + OH' = H 2 0, 

H" + NH 4 OH = H 2 + NH 4 * 

i.e. from the "normal" heat of neutralisation must be subtracted the 
heat necessary to decompose NH 4 OH into the ions NH 4 * and OH'. 
The heat of ionic dissociation for acids and bases is not as a rule great, 
so that in the majority of cases the heat of neutralisation even of 
weak acids and bases does not greatly diverge from the value 137 K. 
The divergence may be in the one direction or the other, according as 
heat is absorbed or developed on the ionic dissociation (cp. p. 252). 
"When acids or bases are so weak that their salts undergo extensive 


hydrolysis in aqueous solution, i.e, are partially split up into free acid 
and base, the heat of neutralisation is very small. This is owing to 
the fact that the neutralisation is incomplete, free hydrogen ions or 
free hydroxyl ions remaining in the solution. 

If we consider the displacement of a weak acid from its salts 
by a strong acid in the light of the dissociation hypothesis, we find 
that the resistance of the weak acid to dissociation is the determining 
circumstance in the reaction. Thus if the salt is sodium acetate and 
the strong acid is hydrochloric acid, the customary equation becomes 

Na' + C 2 H 3 2 ' + H* + CI' = Na* + CV + HC 2 H 3 2 , 

C 2 H 3 2 ' + H' = HC 2 H 3 2 . 

The action is essentially a union of hydrogen ions with acetions, and 
the nature of the salt or of the strong acid is a matter of indifference, 
provided that they are both almost dissociated at the dilution under 
consideration. Similarly, the displacement of a weak base from its 
salts by a highly dissociated base consists essentially in the union of 
a positive ion with a hydroxyl ion, thus 

NH/ + CI 7 + Na + OH' = Na* + Ol 7 + NH 4 OH 

nh; + oh' = nh 4 oh. 

The solution of a metal in an aqueous and strongly dissociated 
acid is principally transference of a positive electric charge from 
hydrogen to the metal: thus if zinc is the metal and hydrochloric acid 
the highly dissociated acid, we have 

Zn + 2H* + 201' = Zn" + 201' + H 2 , 

Zn + 2H* = Zn" + H 2 . 

Similarly, the displacement of bromine from a soluble bromide by 
chlorine is chiefly a transference of an electric charge from bromine to 
chlorine : — 


Cl 2 + 2K' + 2Br' = Br 2 + 2K' + 201' 
Cl 2 + 2Br' = Br 2 + 2Cl'. 

When two highly dissociated salts are brought together in 
solution, we have an ionic equation such as the following, if double 
decomposition is supposed to occur : — 

Na* + 01' + K* + Br' = Na* + Br' + K* + 01'. 

Both sides of this equation are the same, i.e. no chemical change has 
taken place at all This of course only holds good as long as all the 


substances remain in the solution. If the solution is evaporated, that 
salt which is the least soluble will in general fall out first. 

We obtain a similar equation for the action of a strong acid on 
the salt of an equally strong acid, for example — 

H* + CI' + K* + Br' = H* + Br' + K* + CI'. 

Here again both sides of the equation are the same, and thus no 
action has taken place. It is usual to say that in this case the base 
is equally divided between the two acids, but it may be seen from the 
above equation that the ions are free to combine in any way according 
to circumstances. If one of the acids is less dissociated than the 
other, then more of it will exist in the undissociated state, so that the 
other acid is said to have taken the greater share of the base. 

From the examples already given it is evident that the equations 
involving ions are usually of a much more general character than the 
ordinary chemical equations, and more frequently bring out the essential 
phenomenon common to a number of actions of the same type, as, 
for instance, neutralisation, and the displacement of a weak acid or 
base by a stronger. 

The special actions employed in testing for metallic and acid 
radicals are practically always reactions of the ions, so that our 
ordinary tests are tests for ions. Copper, for example, gives a 
black precipitate with hydrogen sulphide, but not under all conditions. 
As long as the copper to be tested for is in the form of a cupric ion 
Cu", the black precipitate is formed when sulphuretted hydrogen is 
introduced into the solution. But if the copper ceases to be a copper 
ion, and becomes part of a more complex ion, the precipitation will 
not take place. This is the case if we add potassium cyanide to the 
solution of a cupric salt until the original cyanide precipitate is dis- 
solved. The copper in the solution is then in the state of the complex 
salt, usually written 2KCN , Cu(CN) 2 . The formula of this salt should 
be written K 2 Cu(CN) 4 , for on solution in water it dissociates electrolyti- 
cally into the positive ions 2K' and the complex negative ion Cu(CN) 4 ". 
The copper is no longer in the form of the cupric ion, but exists merely 
as a part of the complex ion, and has in this state no reactions of its 
own. Sulphuretted hydrogen added to such a solution produces no pre- 
cipitate, and use is made of this fact to separate copper from cadmium. 
Cadmium, it is true, when its salts are treated with excess of potassium 
cyanide, ceases for the most part, like copper, to be a positive ion, 
and enters into the composition of the complex negative ion Cd(CN)/, 
but this ion is not by any means so stable as the corresponding ion 
containing copper. All such compound ions have the tendency to 
split up into simple ions, according to equations like the following : — 

Cd(CN)/ = Cd(CN) 2 + 2CN', 
Cd(CN) 2 = Cd" + 2CN', 


and this tendency exists to different extents with different ions. With 
the complex ion containing cadmium it is very pronounced ; with the 
complex ion containing copper it is much less evident. In the solution 
of K 2 Cu(CN) 4 there is thus a scarcely appreciable quantity of the 
cupric ion Cu", while in the solution of K 2 Cd(CN) 4 the ion Cd" 
exists in moderate proportions. The former solution then is scarcely 
affected by hydrogen sulphide, while the latter is freely precipitated. 
A solution of potassium ferrocyanide, K 4 Fe(CN) 6 is almost entirely 
free from ferrous ions Fe", and exhibits none of the ordinary re- 
actions of ferrous salts. 

The commonest reaction for silver is the production of a precipi- 
tate of silver chloride by the addition to the silver solution of a soluble 
chloride. This reaction is for the silver ion Ag', and not for silver in 
general. If we add potassium cyanide to a silver solution until the 
original precipitate of silver cyanide redissolves, the addition of a 
soluble chloride fails to produce a further precipitate. Here again the 
silver has become part of a fairly stable complex negative ion, Ag(CN) 2 ', 
which gives off very few silver ions by dissociation, so that the 
ordinary reagents for the silver ions may fail to detect their presence. 
The same holds good for the solution of a silver salt in presence of 
sodium thiosulphate. When we add this salt to a solution of silver 
nitrate, we obtain first a white precipitate of silver thiosulphate, 
Ag 2 S 2 3 , which on further addition of sodium thiosulphate dissolves 
with formation of the double salt NaAgS 2 3 . This salt dissociates 
chiefly into the ion Na* and the complex negative ion AgS 2 3 / , so that 
very few silver ions are at any one time in the solution, and the 
ordinary reagents produce none of the characteristic silver reactions. 

A change in the quantity of electricity associated with a positive 
or negative radical is accompanied by an entire change in the pro- 
perties of the radical. Thus the reactions of the ferrous ion Fe" are 
entirely different from the reactions of the ferric ion Fe*" ; and the 
reactions of the ion of the permanganates Mn0 4 ' differ greatly from 
the reactions of the ion of the manganates Mn0 4 ". In connection 
with such changes in the electric charges of ions, the student will find 
it useful to remember that addition of a positive charge or removal of 
a negative charge corresponds to what is generally known as oxida- 
tion in solution ; and that removal of a positive charge or addition of 
a negative charge corresponds to reduction. Thus we are said to 
oxidise a ferrous salt to a ferric salt when we convert the ion Fe" into 
Fe"', or reduce a permanganate to a manganate when we convert the 
ion MnO/ into the ion Mn0 4 ". In the first instance a positive charge 
is removed ; in the second a negative charge. 

If we write the equation 

2FeS0 4 + H 2 S0 4 + Cl 2 = Fe 2 (S0 4 ) 3 + 2HC1, 
on the supposition that all the electrolytes are fully dissociated, we obtain 

2Fe" + 2H* + 3SO/ + Cl 2 = 2Fe"* + 2H' + 3S0 4 " + 2C1' 


2Fe" + Cl 2 =2Fe"' + 2Cl / . 

The whole action, from this point of view, reduces itself to the 
simultaneous appearance of positive and negative charges. The 
ferrous ion assumes a positive charge and is " oxidised " to the ferric 
ion. The uncharged chlorine assumes a negative charge and is 
" reduced " to the chloride ion. In this instance no oxygen has been 
transferred, so that it is only by analogy that we can call such a 
process one of oxidation. It is precisely in these cases, however, that 
the above mode of viewing the action is sometimes of service. When 
actual transference of oxygen takes place, the composition alters as 
well as the charge of the ions, and the ionic conception of the process 
can only be applied to groups of substances which do not change in 
composition. Thus if we consider the action 

10FeSO 4 + 8H 2 S0 4 + 2KMn0 4 = K 2 S0 4 + 2MnS0 4 + 5Fe 2 (S0 4 ) 3 + 

8H 2 

to take place at such an extreme degree of dilution that all the electro- 
lytes are fully dissociated, the equation becomes 

10Fe" + 16H* + 2K' + 18S0 4 " + 2Mn0 4 ' = lOFe"" + 2K* + 2Mn" + 
18SO/ + 8H 2 0; 

lOFe" + 16H" + 2Mn0 4 ' = 10Fe'" + 2Mn" + 8H 2 0. 

Here again we have the " oxidation " of the ferrous to the ferric ion. 
The group 16H' + 2Mn0 4 ' has lost twelve positive and two negative 
charges in becoming the group 2Mn" + 8H 2 0, i.e. has lost on the whole 
ten positive charges, and has therefore been " reduced " by this amount. 
When a metal passes into solution as an ion, it gains one or more 
positive charges, and thus acts as a reducing agent. Thus in the 

Zn + H 2 S0 4 = ZnS0 4 + H 2 ; 

Zn + 2H' = Zn' + H 2 , 

the zinc has been " oxidised " to the state of a positively charged ion, 
while the hydrogen has been " reduced " from the ionic condition to 
the state of free hydrogen. It is scarcely customary to apply 
the terms oxidation and reduction to the passage of hydrogen, or a 
metal from the free state to the state of combination in an acid or 
salt, but the application is obviously justifiable. Free zinc and free 
hydrogen are undoubtedly reducing agents under proper conditions, 
while the ionic zinc or hydrogen, in hydrogen or zinc sulphates, can 
in no sense be looked on as reducing substances in dilute solution. 
Many applications are made in the laboratory and in the operations 


of technical chemistry of the diminution in solubility suffered by a 
salt, acid, or base when there is added to the solution another electro- 
lyte having one ion in common with the original electrolyte (p. 293). 
If we wish to prepare a pure specimen of sodium chloride, we make a 
strong solution of the impure salt, and pass hydrochloric acid gas into 
it, or add to it strong hydrochloric acid solution. The bulk of the 
sodium chloride is precipitated, and in a higher state of purity than 
the salt originally dissolved. Suppose the sodium chloride solution to 
be saturated, or nearly saturated. The addition of the highly disso- 
ciated and extremely soluble hydrogen chloride greatly increases the 
number of chloride ions, and the number of sodium ions must con- 
sequently be diminished by separation of sodium chloride, in order 
that the product of the ionic concentrations shall be maintained 
nearly constant. Even if the solution is not nearly saturated to begin 
with, the addition of chloride ions in sufficient quantity from the 
hydrogen chloride may bring the ionic product up to the constant 
value, and thus determine the precipitation of a portion of the sodium 

If the above process of purification is to be effective, it is essential 
that the added substance should be considerably more soluble than the 
original substance, especially if the two are about equally dissociated, 
as is the case with most salts, strong acids, and strong bases. If we 
attempted to precipitate a saturated solution of the soluble sodium 
chloride by the addition of the comparatively sparingly soluble barium 
chloride, we should find very little sodium chloride to be thrown down. 
This is because the ionic solubility product of barium chloride is much 
smaller than the corresponding product for sodium chloride, owing to 
the smaller solubility and also to some extent to the smaller degree of 
dissociation. The addition of barium chloride contributes therefore 
very few chloride ions to the salt solution, and consequently the 
sodium ions are not greatly reduced in number by precipitation. The 
effect is all the smaller, because the great number of chloride ions from 
the sodium chloride necessitates a very small number of barium ions 
from the barium chloride, in order that the ionic solubility pro- 
duct may not be exceeded, i.e. the solubility of barium chloride 
in saturated salt solution is much lower than in water, so that 
very little of the salt passes into solution to displace the sodium 

The sodium salts of aromatic sulphonic acids are often obtained 
pure from solution by the addition of sodium chloride or caustic soda. 
These salts are, as a rule, much less soluble than either sodium chloride 
or sodium hydroxide, and are therefore thrown out of solution in great 
part when sodium ions from strong brine or solid caustic soda are 

Organic acids, especially those which are not highly dissociated, 
may often be readily precipitated from aqueous solution by the 


addition of hydrogen chloride, the hydrogen ion being here the active 
substance. Even a comparatively soluble and highly dissociated acid 
like sulphocamphylic acid may be thrown out of its aqueous solution 
almost entirely by saturating with gaseous hydrogen chloride. 

The salting out of soap is one of the oldest applications of the 
principle under discussion. "When a fat is saponified with a moderately 
dilute solution of caustic soda, the whole gradually passes into 
solution, and to produce a good soap it is necessary to separate the 
sodium salts of the fatty acids which constitute it from the glycerine 
formed during the saponification and the excess of caustic alkali that 
was employed. The separation can be easily effected by the addition 
of common salt or a strong brine. The sodium salts of the higher 
fatty acids are comparatively slightly soluble in water, and thus the 
addition of a soluble salt like sodium chloride throws them almost 
completely out of solution as a curdy mass. If the fat is saponified 
by a strong solution of caustic soda, there may be enough sodium 
in the form of ions in this solution to throw the soap out as it is 

If a fat is saponified with potash instead of with soda, a solution of 
the potassium salts of the fatty acids is obtained. This solution, if strong 
enough, assumes on cooling a consistency expressed by the name of 
" soft " soap. Soft soaps are not in general salted out as such, but are 
used while still mixed with glycerine and excess of alkali. If we wish 
to obtain the potassium salts free from these admixtures, we may do so 
by adding to the solution a strong solution of potassium chloride. 
The potassium salts of the fatty acids then separate out as a somewhat 
gelatinous mass, containing a considerable quantity of water and still 
retaining the characteristics of a soft soap. 

In the old process of manufacturing hard or soda soaps, the fat 
was saponified with potash, as that alkali was most readily obtainable 
from wood ashes. To the solution obtained on saponification sodium 
chloride was then added. The effect of this was to throw out, not 
a soft potash soap, but a hard soda soap, in virtue of a decomposition 
taking place between the sodium chloride and the potassium salts, 
whereby potassium chloride and the sodium salts were produced. 
These sodium salts were then thrown out by the excess of sodium 
chloride. The addition of sodium chloride to a solution of the potassium 
salts could not to any considerable extent throw them out as such 
from solution, for these substances have no common ion. If we 
consider that potassium ions, sodium ions, chloride ions, and the ions 
of the fatty acids exist simultaneously in the solution, it is evident 
that these free ions may combine in pairs in two ways, the sodium 
which was originally associated with the chlorine having now the 
choice of combining with the ions of the fatty acids. This actually 
occurs, because the ionic solubility product of the sodium salts of these 
acids is much less than the corresponding magnitude for the potassium 


salts, the sodium salts being much less soluble. Under the given con- 
ditions the actual ionic product of the sodium and the negative ions 
exceeds the solubility value, and thus a portion of the sodium salts 
falls out of the solution, the potassium remaining behind mostly as 
potassium chloride. 

If we consider the saturated aqueous solution of any salt, the 
addition of a non-electrolyte will in general affect the solubility to a 
much smaller extent than the addition of an electrolyte containing one 
ion in common with the substance originally dissolved. There will, 
of course, always be some effect, for the solvent is changed by the 
addition of the foreign substance, and any change in the nature of the 
solvent has its effect on the solubility of the substance considered. If 
the substance added is not itself a solvent for the original substance 
the general effect will be slight precipitation from the saturated aqueous 
solution. If we add a small quantity of an electrolyte which contains 
no ion in common with the original electrolyte dissolved, the effect is 
generally to diminish the ionic dissociation product of the latter, and 
thus increase its solubility. This comes about because the addition of 
the second electrolyte produces double decomposition, so that on the 
whole more of the original ions go to form part of undissociated 
molecules. If double salts or acid salts may be formed, of course the 
equilibrium is thereby greatly complicated, and it is impossible to tell 
without further information how the addition of the new substance 
may affect solubilities. 

"The addition of a salt of acetic acid to the acid itself is fre- 
quently carried out in the operations of analytical chemistry in order to 
reduce the strength of the acid, i.e. in order to reduce the concentration 
of hydrogen ions (p. 283). If we take a solution of ferrous sulphate 
and add to it a large excess of sodium acetate, it is comparatively easy 
to precipitate the iron as ferrous sulphide by means of sulphuretted 
hydrogen. If no sodium acetate is added, the precipitation does not 
take place, a dark coloration at most being effected. The old 
explanation of this difference was that by the addition of sodium 
acetate to the ferrous sulphate solution the acid liberated by the 
hydrogen sulphide according to the equation 

FeS0 4 + H 2 S = FeS + H 2 S0 4 

would at once act on the sodium acetate, producing sodium sulphate 
and acetic acid : — 

H 2 S0 4 + 2NaC 2 H 3 2 = 2HC 2 H 3 2 + Na 2 S0 4 . 

The ferrous sulphide was supposed to be easily soluble in sulphuric 
acid, but insoluble in the acetic acid. This explanation is insufficient, 
however, for if we take ferrous sulphate and add to it no more sodium 
acetate than is necessary for double decomposition, the precipitation 
takes place to a slight extent only ; and if we now add acetic acid, the 


precipitate originally formed dissolves up. Acetic acid, therefore, in 
these circumstances dissolves ferrous sulphide. If, however, we now 
add more sodium acetate to the same solution, the ferrous sulphide is 
reprecipitated. The reprecipitation is therefore due to the reduction 
of the degree of dissociation of the acetic acid by the addition of 
sodium acetate, and not to double decomposition alone, as was 
formerly supposed. Ammonium chloride acts in a similar way on a 
solution of ammonium hydroxide, greatly reducing the dissociation 
and therefore the strength of the base. 

It was stated on page 293 that the addition of any strongly dis- 
sociated electrolyte, say sodium chloride, would have practically the 
same effect on the dissociation of acetic acid as the addition of a highly 
dissociated acetate. "Whilst this is true, it must not be supposed 
that the addition of sodium chloride to a solution of a ferrous salt, 
which has enough acetic acid in it just to prevent precipitation by 
sulphuretted hydrogen, will have the same ultimate effect as an 
equivalent quantity of sodium acetate. The degree of dissociation 
of the acetic acid is indeed diminished in the same ratio as before, 
but hydrochloric acid is simultaneously formed by double decomposition, 
and as this is highly dissociated, the number of hydrogen ions in the 
solution, after the addition of the sodium chloride, is rather greater 
than before, so that there is an actual increase in the total active acidity 
of the solution. Precipitation of the sulphide therefore does not occur. 

In analytical chemistry it is a common practice to wash a precipi- 
tate with the fluid precipitant, especially in quantitative operations. 
The theoretical basis of this is, of course, that the sparingly soluble 
precipitate has an ion in common with the precipitant, so that it is 
less soluble in the solution of the latter than in pure water. If other 
circumstances permit, therefore, it is advisable to wash with a diluted 
solution of the precipitant rather than with water alone. 

When solutions of two electrolytes are brought together, and it is 
theoretically possible by double decomposition to obtain a substance 
which is insoluble, then, in general, the double decomposition actually 
takes place. Thus when any sulphate is added to any barium salt, double 
decomposition invariably takes place, barium sulphate being deposited. 
From the point of view of the dissociation theory the explanation is 
at once apparent. The solubility product of the ions of barium sulphate 
is very small, and as soon, therefore, as barium ions and sulphate ions 
are brought together in quantity to exceed this solubility limit, barium 
sulphate fall out. Now all barium salts give barium ions freely in 
solution, and all soluble sulphates behave in like manner. Precipitation 
of b^-ium sulphate therefore invariably occurs, unless, indeed, the solu- 
ti.v^ are so extremely dilute that the solubility product is not reached. 

Exceptions to this rule are only encountered when there is the 
possibility of the existence or formation of complex ions, or when an 
acid or a base is one of the pair of substances. If we add a solution 



of silver sodium thiosulphate to a solution of sodium chloride, there is 
the theoretical possibility of the action 

Na AgS 2 3 + NaCl = Na 2 S 2 3 + AgCl, 

but this action does not occur because there are so few silver ions given 
off by the sodium silver thiosulphate that the solubility product of silver 
chloride is not exceeded, practically all the silver in solution being in 
the form of the complex ion, AgS 2 3 '. For the same reason, where 
cyanides or ammonia are present, there is frequently no precipitation 
where such might be expected by double decomposition. 

If sulphuric acid itself is added to a barium salt, barium sulphate 
is precipitated as readily as if any other sulphate had been employed. 
It is different in the case of tartrates. If calcium chloride is added to 
sodium tartrate solution, a precipitate of calcium tartrate is formed. 
Should tartaric acid, however, be used instead of sodium tartrate, no 
precipitate is produced. The reason for this difference in behaviour 
is not far to seek. When sodium tartrate is employed, tartrate ions 
are abundantly present in the solution, the sodium salt being highly 
dissociated, and the solubility product of calcium tartrate is far 
surpassed. When hydrogen tartrate is employed, we have compara- 
tively few tartrate ions, for tartaric acid is not highly dissociated, and 
their number is still further reduced by the presence of the other 
highly dissociated substances (i.e. the calcium chloride, and the 
comparatively small amounts of hydrogen chloride and calcium 
tartrate formed by the double decomposition), which diminish 
greatly the degree of dissociation of the least dissociated substance, 
viz. tartaric acid. The ionic product of calcium and tartrate ions 
therefore falls short of the solubility product of calcium tartrate, and 
there is no precipitation. For the same reason, some metallic solutions 
are easily precipitated by an alkaline sulphide and not at all by 
sulphuretted hydrogen. In general, it may be said that when all 
the substances concerned are highly dissociated in solution, and 
complex ions are excluded, the precipitation occurs when there is 
the theoretical possibility of it ; while if one of the reacting substances 
is feebly dissociated, the precipitation may only take place to a limited 
extent or not at all. The precipitation may be almost perfect, even 
when a feebly dissociated substance is concerned, if the solubility 
product is vanishingly small, i.e. if the substance is scarcely at all 
soluble. This is the case, for example, with silver sulphide, so that 
sulphuretted hydrogen, although very sparingly dissociated, easily 
precipitates this substance from a solution of silver nitrate. 

In intimate connection with the formation of precipitates on mixing 
electrolytic solutions, there is the solubility of precipitates in. solu- 
tions of electrolytes. It must be borne in mind that the so-ct'^ 'd 
insoluble substances are merely sparingly soluble substances, and 
that the presence of electrolytes in the solvent water only alters the 


solubility by affecting the number of the ions which by their union 
might form the precipitate. Kohlrausch, from measurements of the 
electric conductivity of the saturated solutions, determined the 
solubility of some of the commoner "insoluble" substances, and his 
results are given in the following table, the solubility being expressed 
in parts per million, i.e. milligrams per litre, at 18° : — 

Solubility. 1 

Silver chloride 1 *7 

Silver bromide 0*4 

Silver iodide ....... 0*1 

Mercurous chloride ...... 3'1 

Mercuric iodide ...... 0*5 

Calcium fluoride . . . . . . 14 

Barium sulphate . . . . .' . 2*6 

Strontium sulphate 107 

* Calcium sulphate 2070 

Lead sulphate . 46 

Barium oxalate . . . . . . 74 

Strontium oxalate ...... 45 

Calcium oxalate 5*9 

Barium carbonate 24 

Strontium carbonate 11 

Calcium carbonate 13 

v Lead carbonate 3 

Silver chromate 28 

Barium chromate 3*8 

Lead chromate ...... 0*2 

Magnesium hydroxide ..... 9 

The solubility product of silver chloride in water is very small, 
corresponding to the very slight solubility of the salt. If we have 
the solid salt in presence of water, and add nitric acid to the solution, 
we disturb the equilibrium very little. The dissolved silver chloride 
is almost entirely dissociated, and the silver nitrate and hydrogen 
chloride, which might be formed from it by the action of the nitric 
acid, would be likewise almost entirely dissociated. The addition of 
nitric acid, therefore, does not appreciably affect the silver and chloride 
ions in the solution, and therefore is without influence on the solubility 
of the silver chloride. Consider, on the other hand, the effect of the 
addition of hydrochloric acid on the solubility of calcium tartrate. 
The calcium tartrate in the aqueous solution is highly dissociated, 
but the addition of hydrochloric acid at once liberates tartaric acid, 
which is only slightly dissociated in the presence of the highly 
dissociated calcium chloride, etc. The concentration of the tartrate 
ions is therefore much reduced, and the ionic product falls below the 
solubility product, i.e. the solution becomes unsaturated with respect 
to calcium tartrate. More of this salt, therefore, dissolves up, and the 
process of solution goes on until the ionic product once more reaches 

1 These solubilities are only approximate, and are probably too great for the least 
soluble salts. Thus it has been found by another electrical method that the solubilities 
of silver chloride, bromide, and iodide, at 25°, are respectively 1*8, 0*13, and 0*0017 mg. 
per litre. 


the solubility product. It is evident from these considerations that a 
given quantity of hydrochloric acid will not dissolve an unlimited 
amount of calcium tartrate. If, however, we take a sufficient quantity 
of acid, a given amount of calcium tartrate can always be entirely 

In general we may say that strong acids in aqueous solution will 
not appreciably dissolve salts of equally strong acids, or even of acids 
nearly as strong, for any double decomposition that takes place in 
solution does not then appreciably affect the number of the ions 
which regulate the solubility. On the other hand, strong acids will 
in general easily dissolve insoluble salts of weak acids, for then the 
weak acid is liberated, which being feebly dissociated, reduces the ionic 
product, with the result that the solid dissolves to restore the solution 
equilibrium. Sometimes, when the solubility product of the salt of 
the weak acid is excessively small, as it is in the case of silver 
sulphide, even an equivalent of a strong acid like nitric acid, will not 
dissolve up an appreciable quantity, for the solubility product is soon 
reached when silver nitrate and hydrogen sulphide begin to accumulate 
in the solution. Weak acids, as we might expect, do not dissolve the 
" insoluble " salts of stronger acids. Whilst calcium oxalate is freely 
soluble in hydrochloric acid, it is almost insoluble in acetic acid. In 
the first case the number of oxalate ions in the aqueous solution is 
diminished by the addition of the hydrochloric acid, whereas in the 
second it is not sensibly affected. Calcium oxalate, therefore, must 
dissolve in hydrochloric acid solution to restore the solubility product. 
When acetic acid solution is added, the ionic product scarcely departs 
from the solubility value, so that no calcium oxalate need pass into 

As has already been indicated, the number of ions of a given kind 
in a solution may be greatly altered by the formation of complex ions. 
If to a saturated solution of silver chloride in contact with the solid 
there is added a quantity of potassium cyanide, the free cyanide ions 
unite with a portion of the silver ions to form complex ions AgC 2 N 2 '. 
The ionic product of silver and cyanide ions, therefore, falls below the 
equilibrium value, i.e. below the solubility product, with the result 
that silver chloride must dissolve up in order to restore equilibrium. 
If the quantity of cyanide added is small, the silver chloride need not 
dissolve wholly, but if a sufficient excess of cyanide is employed, the 
solution of the silver chloride will be complete. Should the solubility 
product be extremely small, as is the case with silver sulphide, potas- 
sium cyanide has only a slight solvent action, and the silver sulphide 
dissolved is easily reprecipitated on addition of potassium sulphide. 
The solvent action of sodium thiosulphate or ammonia on " insoluble " 
silver compounds is similar in origin, the increased solubility being 
due to the formation of complex ions with corresponding disappear- 
ance of silver ions. It will be noticed that the solubility of silver 


chloride, bromide, and iodide in water (given in the table on p. 307), 
is the same as the order of their solubility in ammonia, as an applica- 
tion of the above theory would lead us to expect. 

The theory of electrolytic dissociation affords some assistance in 
understanding the action of the indicators used in acidimetry and 
alkalimetry. The indicators are themselves acid or alkaline in nature, 
but are necessarily very feeble compared with the acids or alkalies 
whose presence they indicate. Their action depends on a change of 
colour which they undergo on neutralisation. Phenol phthaleine, for 
example, is a substance of very weak acid character, being in aqueous 
solution almost entirely undissociated. If, however, we add a strong 
alkali such as sodium hydroxide to it, the sodium salt is formed and 
imparts an intense pink colour to the solution. All the soluble salts 
of phenol phthaleine are thus coloured, and the colour is of the same 
intensity in equivalent solutions if these solutions are very dilute, 
so that we are justified in concluding in terms of the dissociation 
hypothesis, that the colour is due to the negative ion of the phenol 
phthaleine, as this is the substance common to all dilute solutions 
of salts of phenol phthaleine. The undissociated substance has no 
colour. Let us consider what happens as we titrate a solution of an 
acid, using phenol phthaleine as an indicator. In presence of the 
acid, the indicator, being a very feeble acid, is even less dissociated 
than it would be in pure water, and consequently no colour is per- 
ceptible. As soon, however, as the acid originally present in the 
solution is neutralised and a drop of alkali in excess is added, the 
corresponding alkaline salt of phenol phthaleine is formed, i.e. negative 
ions from the phenol phthaleine are produced, and the solution at 
once assumes the pink tint characteristic of them. In order to have 
a sharp indication of the neutral point, it is necessary first that the 
acid to be titrated should be considerably stronger than phenol 
phthaleine itself, and that the alkali should be a strong alkali. If the 
acid to be titrated is such a feeble acid that its salts even with strong 
bases suffer hydrolysis in aqueous solution, it is obvious that phenol 
phthaleine is incapable of sharply indicating the neutral point, for 
long before sufficient alkali for complete neutralisation has been added, 
the salt formed will begin to split up into free acid and free base, part 
of which will neutralise the phenol phthaleine, and thus produce a faint 
pink colour which will gradually deepen in intensity as the addition of 
alkali progresses. Carbolic acid, and other phenols, therefore, cannot be 
titrated with alkali and phenol phthaleine, on account of the hydro- 
lysis which their salts suffer (cp. p. 278). Polybasic acids, too, very 
frequently form normal salts which are partially hydrolysed in 
aqueous solution, and with them also no definite indication of the 
neutral point can be obtained. Thus the ordinary sodium phosphate, 
i.e. di-sodium hydrogen phosphate, although formally an acid salt, has 
an alkaline reaction to phenol phthaleine on account of its slight 


hydrolysis. Phenol phthaleine roughly indicates neutrality in the 
case of carbonic acid when sodium hydrogen carbonate exists in the 
solution, and gives a strong pink colour with the normal sodium 
carbonate. Since carbonic acid thus behaves as an acid to phenol 
phthaleine, i.e. since carbonic acid is a stronger acid than phenol 
phthaleine, it must be excluded from the alkali with which acids 
are titrated. This is best done by using baryta as the alkali, and 
keeping it protected from the carbonic acid of the atmosphere. Any 
carbonic acid which may have been originally present settles down as 
barium carbonate, and the clear liquid is therefore free from this 
source of disturbance. 

If the base employed is not a strong base, the indication in this 
case also is uncertain owing to hydrolysis. Thus ammonia should 
never be used in titrations with phenol phthaleine as indicator, for 
the ammonium salt of phenol phthaleine, being the product of the 
union of a weak base with a very weak acid, is hydrolysed in aqueous 
solution, and thus the neutrality point is not sharply indicated, even 
though the salt of ammonia with the acid originally in the solution 
undergoes no hydrolysis. 

Phenol phthaleine then forms a good indicator when weak acids 
are titrated with strong bases. 

The application of the dissociation theory to explain the action of 
other indicators is not so simple as in the case of phenol phthaleine, 
because these indicators are usually of mixed function, being them- 
selves acidic or basic according to circumstances, and thus forming two 
series of salts, one series with strong acids, and another with strong 
bases. There may, therefore, be more than one kind of coloured ion 
in solution, besides, perhaps, coloured undissociated indicator, so that 
the interpretation of the actual colours of the different solutions on 
the dissociation theory is somewhat arbitrary. 

Many applications of the electrolytic dissociation theory to ordinary 
laboratory work, will be found in 

W. Ostwald, The Scientific Foundations of Analytical Chemistry. 



The experience of practical and scientific men alike goes to show that 
it is impossible to construct a perpetual motion machine. In the 
general acceptance of the term, a perpetual motion machine is one 
from which more energy can be obtained than is put into it from the 
outside. This may be termed a perpetual motion of the first class. 
But another kind of perpetual motion machine might exist, one 
namely, which could by a recurring series of processes continuously 
afford mechanical energy at the expense of the heat of surrounding 
bodies at the same temperature as itself. This kind of perpetual 
motion is also impossible, and has been termed perpetual motion of the 
second class. It must be clearly understood that not only have such 
machines never been constructed, but that no increase in our experimental 
skill at present conceivable could lead to their construction. If there- 
fore in argument we find that an imaginary series of processes would 
lead to a perpetual motion of either of the kinds mentioned above, 
we conclude that such a series of processes can have no real existence. 
The denial of the possibility of the existence of the two sorts of 
perpetual motion is contained in the two following positive and general 
statements, which are known as the First and Second Laws of 
Thermodynamics, respectively. 

I. The energy of an isolated system remains constant. 

II. The entropy of an isolated system tends to increase. 

With the second law in its general and formal aspect we shall 
have here little to do, and shall use in its stead the negative proposi- 
tion given above. Entropy is a function, which while theoretically of 
great value as indicating the direction in which chemical or other 
processes take place, and in fixing generally the conditions of equili- 
brium, is not susceptible of direct measurement, and is consequently 
of less obvious and immediate practical importance. 

The first law states the principle of the Conservation of Energy, by 
an isolated system being meant a system which can neither give up 
energy to its environment nor absorb energy from it. The sum of 


the different kinds of energy in such a system is always the same, no 
matter what forms the energy may assume. In order to ascertain 
practically that the sum of the energies is constant, we must obviously 
have one kind of unit in which all energies may be expressed. If we 
work with only one kind of energy, we express its amount in the 
appropriate unit. Thus we express amount of heat in calories ; 
electrical energy in volt-coulombs, or the like ; and mechanical energy 
in foot-pounds or gram-centimetres. In dealing with different kinds 
of energy, we may take any of these units as the standard in which 
we express all the different varieties, for we know the factors necessary 
for converting one unit into any of the others. One calorie, for 
example, is under all circumstances equivalent to 42,000 gram- 
centimetres, so that if we wish to add heat energy and mechanical 
energy together, we must either multiply each calorie of heat energy 
by 42,000 to convert it into gram-centimetres, or divide the number 
of gram -centimetres by the same number in order to find their 
equivalent in calories. A few simple deductions from the First Law, 
involving only the consideration of thermal and mechanical energy, 
will now be given. 

When a small quantity of heat dQ is supplied to a gram-molecular 
weight of a perfect gas at constant volume, the heat goes entirely to raise 
the temperature of the gas, and we may therefore write dQ = C^dT ; where 
C v is the heat capacity of the gram molecule at constant volume (cp. 
p. 34), and dT the small rise of temperature which this amount of 
gas experiences. If the heat is supplied to the gas under constant 
pressure, it not only goes to raise the temperature, but is also partially 
converted into work done by the gas during expansion. If p is the 
constant pressure, and dv the change of volume, this work is repre- 
sented by the product pdv. Now according to the First Law 

dQ = C v dT+pdv 9 (1) 

if no other kind of energy is involved, and if the heat and mechanical 
energy are expressed in terms of the same unit. 

In the gas equation for the gram molecule (pv = BT), p, v, and T 
are variable, so that on differentiation we obtain 

pdv + vdp = BdT. (2) 

By eliminating dT from equations (1) and (2) we obtain 

dQ = -~—pdv + ~vdp ; 

or since C p = C v + B, as was shown on p. 34, 

dQ = -gpclv + -£vdp. 

Now, if the pressure, volume, and temperature of the gas be 


allowed to change adiabatically, i.e. in such a way that heat neither 
enters nor leaves the system, we have dQ = 0, and consequently 

C p pdv + C v vdp = 0. 

Introducing into this equation the ratio of specific heats k = C p /C w we 

kpdv + vdp = ; 

v p 
This equation, when integrated between the values ^w and^?^, gives 
k(\og e v 1 - log e v) + log e p 1 - log e p = ; 

whence k = 1 0&P "^: 

log^ - l0g c V 



These results are of importance as they enable us to ascertain the 
ratio of the specific heats of gases by observations of pressures and 
volumes under circumstances in which heat can neither leave nor 
enter the gas considered. For example, during the rapid compressions 
and dilatations which constitute the passage of sound through a gas, 
there is no time for the heat change at any portion of the gas to com- 
municate itself by conduction to neighbouring portions, so that the 
process instead of being isothermal, is, so far as any given portion of 
the gas is concerned, an adiabatic process, the gas being cooled at each 
dilatation and warmed at each compression. The gas then does not in 
these circumstances obey Boyle's Law, which holds good only for 
isothermal change of pressure and volume, but the law given above, 
where the pressure is not inversely proportional to the volume, but to 
the k-ih power of the volume. It is owing to this circumstance that 
we can deduce the ratio of specific heats of a gas from the speed at 
which sound travels in it. 

We have now to find the law connecting the volume of a gas with 
its absolute temperature when it is heated adiabatically, i.e. by com- 
pression, since no heat as such must enter the system. This can 
easily be done by eliminating pdv from equations (1) and (2), the 
result being 

dQ = (C v + E)dT-vdp 

= C p dT-vdp 

= C p dT-^-BT, 
P P 
since pv = RT. 

For an adiabatic compression dQ = 0, so that 


CJT - ^-BT=0, 
P P 


= 0; 

C p dT 


k dT 


k-l T 


Integrating between T, p, and T v p v we obtain 

j— j(lo&ri - log, T) - (\og e p 1 - log e p) = 0, 

But since in an adiabatic process ±± = f — ) , we obtain finally 

%'- fet" 




This result we shall find useful in calculating the maximum amount of 
work which can be obtained from a given quantity of heat under 
given conditions, a problem which we shall now proceed to solve. 

According to our statement of the second law, no work can be 
obtained from the heat contained in a number of bodies all having the 
common temperature of their surroundings. In order to convert heat 
into work, we must have temperature differences. A body in cooling 
through a certain range of temperature parts with a certain amount 
of heat, and a definite fraction of that heat may be transformed into 
work. In actual practice, different engines will effect the conversion 
of different amounts of the heat, but there is a theoretical limit which 
no engine, however perfect, can exceed, and it is our task to find what 
that limit is. 

The process of reasoning is essentially the same as that of Carnot, 
who introduced the conceptions necessary for the solution of the 
problem, and arrived at the desired result, although to him heat was 
a material substance, and not, as we now believe, a form of energy. 
The fundamental conception is that of a reversible cycle of opera- 
tions. By a cycle of operations is understood a series of processes 
which leaves the system considered in exactly the same state as it was 
initially, and the term reversible applies not merely to the direction 
of the mechanical operation, but to the complete physical reversibility 
of all the processes involved. 


A reversible heat-engine after converting a certain amount of heat 
into work will return to its original state in every respect if made to 
act backwards step by step, so that the work obtained is entirely 
converted into heat by its agency. Such an engine is of the maximum 
possible efficiency, for if any engine were more perfect a perpetual 
motion could be obtained. This may be shown as follows. Let A be 
a reversible engine, and let B be an engine which under the same 
conditions can convert a larger proportion of heat into work than A. 
Let the two engines work between the same temperatures, and sup- 
pose that when a quantity of heat Q is given to A, the proportion q is 
converted into mechanical work. If the work corresponding to q is 
now done upon this engine so that the processes are reversed, the 
system will arrive exactly at its initial state, the quantity of heat Q-q 
being raised from the lower to the higher temperature. Now instead 
of letting the engine A act directly, take in its stead the more perfect 
engine B, and supply it at the higher temperature with the quantity 
of heat Q. A greater proportion of this than before is converted into 
work, say q', the quantity of heat Q-q falling to the lower tempera- 
ture. By using the reversible engine to perform the reverse trans- 
formation of work into heat, we can regain the original quantity of 
heat Q at the original temperature, by expending mechanical energy 
equivalent to q } the quantity of heat Q-q being at the same time 
raised to the higher temperature. In the whole series of operations, 
then, the heat q' - q has been taken in at the lower temperature and an 
equivalent amount of work has been gained. This cycle of operations 
can be repeated as often as we choose, so that here we should have a 
system capable of giving an indefinitely large amount of work at the 
expense of heat at the lower temperature, which might be the uniform 
temperature of the surroundings, i.e., we should have realised a 
perpetual motion of the second class. We conclude, then, that an engine 
more perfect than the reversible engine A cannot exist. It will be 
noticed that nothing is said as to the nature of the working sub- 
stances in the reversible or the other engine, so that the conclusion is 
perfectly general. We are at liberty therefore to use any kind of 
reversible engine in our calculations in order to ascertain the maximum 
quantity of heat which can be converted into work, and we shall find 
it convenient to take for our working substance a perfect gas, on 
account of the simplicity of the laws which it obeys. 

We must first of all investigate if the processes through which we 
put the working substance are really reversible. The condition 
of reversibility is that the state of the system at any time does 
not differ sensibly from equilibrium, for then the slightest variation 
in the conditions will determine the occurrence of the process in 
the one direction or the other. If, therefore, we communicate heat 
to the gas, we must so arrange that the gas and the heat source 
have temperatures differing from each other by an infinitely small 


amount. Similarly, if the gas is to part with heat, it must do so 
to a heat-sink with a temperature lower than its own only by an 
infinitesimal quantity. If the gas is to be compressed, the external 
pressure at any instant must be only greater by an infinitely small 
amount than the pressure of the gas itself; and if the gas is to be 
expanded, the external pressure must be less than the pressure of the 
gas by an infinitesimal difference. The machine of course must be 
absolutely frictionless, for otherwise some work would have to be 
expended in moving the machine, and thus converted into heat, inde- 
pendent of the gaseous working substance which is alone considered. 
From all this it is obvious that a reversible engine is an engine which 
can never be realised in practice. For an engine to be strictly 
reversible, there should be no departure from the conditions of 
equilibrium at any stage, in which case no process could occur at all, 
for the occurrence of any process naturally involves a departure from 
equilibrium. If the departure from the equilibrium conditions were 
infinitely small, the process would occupy an infinitely long time in 
its performance. We see then that a reversible process is an ideal, 
just as a perfect gas is an ideal. Neither can ever be met with in 
practice, but this in no way impairs the value of the theoretical con- 
clusions deduced by their aid. 

The series of operations which we shall perform on the gas will be 
best seen in the pressure-volume diagram, Fig. 45. We begin with 

the gas in the state represented in the 
diagram by the point 1. At the con- 
stant temperature T we let the gas 
slowly expand until its pressure and 
volume are indicated by the point 2. 
The form of the curve obtained during 
the expansion is the rectangular hyper- 
bola of gases (cp. p. 76). We next 

ik & a p y ~~ isolate the gas from the heat source of 
FlG . 45< constant temperature T, and let the 

expansion continue adiabatically until 
the point 3 is reached. Since the pressure varies more rapidly 
with the volume in an adiabatic than' in an isothermal process (p. 313), 
the line 2, 3 will be more inclined to the volume axis, than 1, 2, as 
is shown in the diagram. As no heat can enter the system during 
the adiabatic expansion, the temperature will fall, say to T'. We now 
bring the gas into contact with a heat reservoir at the temperature T\ 
and compress it isothermally until a point 4 is reached, such that when 
the compression is continued adiabatically, the adiabatic curve will 
pass through the initial point 1, where the process is stopped and the 
cycle thus completed. 

In this cycle a certain quantity Q of heat has been absorbed by 
the gas at the higher temperature T, and the quantity Q' has been 


given out by the gas at the lower temperature T. At the same time 
a certain amount of work has on the whole been performed. The gas 
on expanding does work, and this work is measured by the product 
of each pressure into the corresponding change of volume. In the 
diagram, therefore, the work performed by the gas on expansion is 
measured by the area a, 1, 2, 3, y. When the gas was compressed, 
work was done upon it, and this work in the diagram appears as the 
area y, 3, 4, 1, a. The total work obtained then from the gas during 
the cycle is the difference of these areas, viz., the quadrilateral 1, 2, 3, 4. 
To obtain actual numerical relations we may consider a gram 
molecule of the gas, for which the previous equations of this chapter 
are valid. During the isothermal expansion 1, 2, the gas absorbed Q 
units of heat at the temperature T, while it expanded from the volume 
v x to the volume v 2 . In equation (1) p. 312. 

dQ^C^T + pdv 

we may put dT equal to zero, since we are considering an isothermal 

process, and for p we may substitute — . We thus obtain 

dQ = Bl^, (la) 

which when integrated between the limits v 1 and «? 2 , gives 

Q = BT log, } 

as the amount of heat absorbed by the gas. 

Similarly for the isothermal compression 3, 4 we obtain 

or Q' = RT\og e J. 

The sign of Q is in the first equation of this pair different from that of 
Q above, since in the first case the system gains heat, and in the second 
loses it. Division now gives 


For the adiabatic expansion 2, 3, we have (p. 314) 

r W 

c/ 2 / 

and for the adiabatic compression 4, 1 



t W 

& = ?§ or ^ = - 4 . 

Equation (3) therefore reduces to 

Q r 

i.e. the heat absorbed is to the heat given out as the absolute tempera- 
ture of the absorption is to the absolute temperature at which the heat 
is lost by the system. A slight alteration gives the equation in the 

Q-Q' T-T 



i.e. the proportion of the absorbed heat which is converted into work 
is equal to the temperature difference between the two isothermal 
operations divided by the temperature of absorption. A form which 
we shall find useful in subsequent calculations is 





These results although derived from a consideration of the be- 
haviour of gases, are valid for all reversible cycles, and can therefore be 

applied in every case 
for which we can show 
all the operations in- 
volved to be revers- 

The first applica- 
tion of equation (4) 
will be to the process 
of vaporisation. Let 
us consider a quantity 
of liquid under a pres- 
sure P which is equal 
to the vapour pressure 
of the liquid at the 
constant temperature 
chosen. An infinite- 
simal diminution of 
the pressure on the 
liquid will cause it gradually to pass into vapour if this pressure 
and the constant temperature are maintained. Suppose one 



Fig. 46. 


gram -molecular weight of vapour to be produced in this way, and 
let the isothermal process be represented in the indicator diagram, 
(Fig. 46), by the line 1, 2 which is parallel to the axis of volumes, 
the pressure being constant. Now expand the vapour adiabatically 
from the original pressure P to a pressure which is dP lower, the 
temperature at the same time falling. The pressure and volume may 
then be represented by the point 3. At a temperature T-dT for 
which the vapour pressure of the substance is P - dP y compress the 
vapour isothermally to liquid and finish the compression adiabatically 
until the original pressure, temperature, and volume are regained. 
The work done upon the system is equal to the area of the figure 1, 2, 
3, 4, which is practically the product of the line 1, 2 into the vertical 
distance between 1, 2 and 3, 4. Now the line 1, 2 in the diagram 
represents the difference in volume between the liquid and the vapour, 
and the distance between the two horizontal lines represents the differ- 
ence of vapour pressure dP due to a difference of temperature dT. 
For the work done then we have the product dP(V-v), where V is 
the molecular volume of the vapour, and v the molecular volume of the 
liquid. The quantity of heat absorbed at the higher temperature T is 
Q, the molecular heat of vaporisation of the liquid at this temperature. 
The quantity which has been transformed into work is consequently 

Q— according to equation (4). If we express the heat in mechanical 

units, as we can do by multiplying the heat units by /= 42,350 (see 
p. 6) we obtain the equation 

dP(V-v) = J<f-±, (5) 

a result of considerable practical importance and of wide applicability. 
As the volume of a liquid is only a fraction of a per cent of the 
volume of vapour derived from it at ordinary pressures, it is often per- 
missible to write simply V instead of V-% which brings about a 
numerical simplification. For the gram-molecular volume of a gas we 
have PV=PiT, where R is equal to 2 J very nearly (see p. 29) so 

that V= -p-. We may therefore write equation (5) in the form 

% ' e * PdT~ 2 r T 2 ^ ' 

or dT - 2T2 (<) 

We may calculate from this result in the form of equation (6) the 
latent heat of vaporisation of benzene from the change of its vapour pres- 


sure with the temperature. At 5° the vapour pressure of benzene is 
34*93 mm. of mercury, or 47*50 g. per sq. cm. ; at 5*58° the pressure 
is 36*06 mm. or 49*04 g. per sq. cm. For dP then we have 1*54, for 
dT we have 0*58, for P we have the mean value 48*27, and for T the 
mean value 278*3. The value of Q is therefore 

2T*dP 2(278*3) 2 xl*54 ■ 
-Pdf~= 48*27x0*58 = 852Qcal - 

The value of the heat of vaporisation actually found is 8420 cal, so 
that the agreement between calculation and experiment is fairly close, 
the difference not being greater than the error of experiment. 

Equation (5) not only holds good for the liquid and gaseous phases, 
i.e. for vaporisation, but also for the equilibrium between any other 
pair of phases, for example, solid and liquid, or two solid phases such 
as the different crystalline modifications of sulphur. In the form 

a.^i, (8) , 

it is useful for ascertaining the effect of pressure on the temperature 
of equilibrium. V, v, and q may all refer either to molecular quantities 
or to unit weight of the substance considered, as the molecular factor 
cuts out in the right-hand member. If we consider the transformation 
brought about by supplying heat to the substance, q is a positive 
quantity, while /and T also are necessarily positive. Consequently dP 
will have the same sign as dT, or the opposite sign, according as V— v is 
positive or negative, i.e. if the substance expands on being transformed 
by application of heat, dP and dT will have the same sign, whilst if it 
contracts dP and dT will have different signs. In the first case then 
the transition point will be raised by increase of pressure; in the 
second case it will be lowered. Rhombic sulphur on melting expands ; 
V - v is therefore positive and the melting point of rhombic sulphur is 
raised by pressure. Again rhombic sulphur expands on passing into 
monoclinic sulphur, and consequently the transition point is raised by 
application of pressure. Water, on the other hand, occupies a smaller 
volume than the ice from which it is produced ; V-v is therefore 
negative, and increase of pressure lowers the melting point. 

As a numerical example of the application of formula (8), we may 
calculate the effect of one atmosphere increase of pressure on the 
melting point of ice. A cubic centimetre of water at 0° is obtained 
from 1*09 cc. of ice; the change of volume on liquefaction is therefore 
0*09 cc. per gram of water. The latent heat of liquefaction per gram 
is 80 cal., and the temperature of liquefaction is T= 273. We have, 
therefore, if dP= 1 atm. = 1033 g. per sq. cm., 

Jq 42350 x 80 


that is, the melting point of ice is lowered 0*0075° for each atmo- 
sphere increase of pressure. A corresponding diminution of pressure 
causes the same rise in the melting point. Thus ice which melts 
under atmospheric pressure at 0°, melts at 0*0075° under the pressure 
of its own vapour, so that the triple point (p. 99) lies at this tempera- 
ture and not at 0°. 

Dilute Solutions 

When we dissolve a substance in any liquid, the process is not, 
under ordinary conditions, a reversible one, for we have not in general 
during dissolution a state bordering on equilibrium. In certain 
circumstances, however, it is possible to conduct the process reversibly. 
If we are dealing, for example, with the solution of a gas in a non- 
volatile liquid, we can proceed reversibly as follows. Let the gas and 
liquid be taken in such proportions that the gas will just dissolve in 
the liquid at the pressure p and the constant temperature of experi- 
ment t. Suppose the liquid and gas to be contained in a cylinder 
with a movable gas-tight piston. At first let there be a partition 
separating the gas and the liquid. Without removing this partition, 
expand the gas by gradually diminishing the pressure in such a way 
that at no instant during the expansion the condition of the gas 
differs sensibly from equilibrium. According to Henry's Law, the 
quantity of gas dissolved by the liquid is proportional to the pressure 
of the gas. Let the expansion be continued until the pressure of the 
gas is so small that practically none of it dissolves in the liquid when 
the separating partition is removed. After removal of the partition 
let the pressure on the gas be increased by insensible gradations. At 
no time does the state of the system deviate sensibly from equilibrium, 
and the pressure may be gradually raised until at the pressure p all 
the gas has dissolved. The solution of a gas in a liquid then may be 
made part of a reversible cycle if the process is carried out as here 

The concentration of a solution of a non-volatile substance in any 
liquid can be changed reversibly by bringing the solution into contact 
with the solvent under equilibrium conditions, and then by an 
infinitely small alteration of the conditions determine a process of 
concentration or dilution. It is apparent at once that we cannot bring 
the solution into direct contact with the liquid solvent, either by 
mixing directly or allowing the dissolved substance to diffuse slowly 
into a fresh quantity of solvent as in the experiment described on p. 
149, for no slight change in the conditions can at any stage make the 
action proceed in the reverse direction, i.e., make a solution separate 
into a more concentrated solution and the solvent. If the solvent is 
in the form of vapour or solid, however, the dilution or concentration 
may be effected reversibly. Suppose the solution to be in presence of 



its own saturated vapour at the constant temperature of experiment. 
An increase of external pressure, however slight, will cause part of 
the vapour to condense, i.e. will dilute the solution ; while a slight 
diminution of the external pressure will cause part of the solvent to 
evaporate, i.e. will concentrate the solution. Similarly, if the solution 
is in equilibrium with the solid solvent, a very slight rise of tempera- 
ture will bring about partial liquefaction of the solid solvent, and thus 
dilute the solution, and a correspondingly slight diminution of tempera- 
ture will produce partial separation of solid solvent and thus con- 
centrate the solution. 

There is still another way of bringing the solution into contact 
with the pure solvent under equilibrium conditions, namely through a 
diaphragm which is permeable to the solvent and not to the dissolved 
substance. If the solution is enclosed in a cylinder with a semi- 
permeable end and a movable piston, it will be in equilibrium with 
the liquid solvent through the diaphragm when there is a certain 
pressure on the piston, — the osmotic pressure. If the pressure of the 
piston is increased ever so slightly, solvent flows outward through the 
semipermeable diaphragm and the solution becomes more concentrated ; 
if the pressure on the piston is diminished, solvent flows inwards 
through the diaphragm, and the solution is thereby diluted. 

All these methods of changing the concentration of a solution can 
therefore be adopted as parts of reversible cycles of operations, and 
we shall see that by removing a portion of solvent from a solution by 
one method, and by adding it to the solution again by another method, 
we obtain a series of results which are of great theoretical and 
practical importance. 

In the first place we shall consider a cycle in which a gas is dis- 
solved in a non- volatile liquid by the reversible process given on p. 
321, and the system then brought back to its original condition by 
means of semipermeable diaphragms. We start with a volume v of 
the gas under pressure^, and with a volume Voi liquid just sufficient 
to dissolve the gas under this pressure, and we propose to find 
what amount of work (positive or negative) must be done in order 
to bring the gas into solution reversibly at constant temperature, 
puring the first stage contact between gas and liquid is prevented by 
a partition inserted at the surface of the liquid. If the cylinder in 
which the gas and liquid are contained have unit cross section, and 
the initial distance of the piston from the liquid surface is x , we have 
for this state x = v . At any stage of the expansion (x) the pressure 

p is given by the equation ^?=^-°, and the work done by the gas 


during the expansion is represented by the expression 


i QiX X 

J V X ^0 




x being a very large multiple of v . The partition is then removed, 
and the pressure on the gas increased. The pressure on the piston in 
a given position x is less than before, for the gas which was previously 
confined to the space x is now partly in solution. If s denote the 
solubility (p. 59), the available volume is practically increased in the 
ratio x : x + s F, so that the pressure in position x is now given by 

p = 


x + s r 

and the work required to be done during the compression is 


x + sF 

=2>oV log< 

x + sF 

On the whole, the work done on the system during the double opera- 
tion is 

J, x + sF 

>o j l0g e - 



?<Po | !og< 

x + sF 

~ loge 


The quantity within the brackets of the second expression may be 




seen to be zero, since x is indefinitely great, so that = 1, and since 


by supposition the quantity of liquid is just 
capable of dissolving the gas, whence sF—v . 
The conclusion, then, is that there is no gain 
or loss of work in dissolving the gas reversibly 
in the liquid. 

The gas may now be removed from solu- 
tion and restored to its original state rever- 
sibly by means of semipermeable membranes 
arranged as in Fig. 47. One membrane gg 
permeable to gas but not to liquid, is intro- 
duced at the surface of the liquid, on which 
the piston KK rests at the commencement 
of the operation. A second membrane 11, 
permeable to liquid but not to gas, is sub- 
stituted as a piston for the bottom of the 
cylinder, and is backed upon its lower side by 
the pure solvent. By suitable proportional 
movements of the two pistons, KK being 
raised through the space v 0i while 11 is raised 
through the space F y the gas may be expelled, 
the pressure of the gas retaining the constant value p , and the solution 
which remains retaining a constant strength, and therefore a constant 





osmotic pressure P. When the expulsion is complete, i.e., when the 

two semipermeable membranes have come together, the work done 

upon the lower piston is PV, and the work done by the gas in raising 

the upper piston is p v . 

The system is now in its original state, and all the operations have 

been conducted reversibly. A reversible cycle has therefore been 

performed, and the equation Q - Q' = — — — Q is applicable. The 

temperature has remained constant throughout, so that T— T' = 0, 
and therefore Q - Q — 0, as in all reversible isothermal cycles. Since 
no heat has been converted into work, or vice versa, the work done on 
the system must on the whole be equal to the work done by the 
system. In the first stage, i.e. the process of solution, it has been shown 
that there is neither loss nor gain of work, so that for the second 
stage we must have 

If the gas which occupies the volume v at the pressure p 0i is made to 
occupy the volume V, its pressure will assume a new value jp, and 
according to Boyle's Law we shall hsivep v =pF ; combining this with 
the previous equation, we then obtain pF-PF and p = P. The 
osmotic pressure then of the gas in solution is equal to the gaseous 
pressure which it would exert in absence of the solvent if it occupied 
the same space at the same temperature, a result in harmony with the 
calculation from experiment made on p. 164. 1 

The above result holds good only for ideal substances and for very 
dilute solutions, since in its deduction we have assumed that the gas is 
a perfect gas which obeys the laws of Boyle and Henry exactly, and 
that the volume of the solution is exactly the same as the volume of 
the solvent which it contains, an assumption which can only be justifi- 
ably made when the solution is extremely dilute. The solvent, too, 
was supposed to be non- volatile, and the reversible processes themselves 
are purely ideal. Notwithstanding all these assumptions the conclusion 
arrived at is practically important, and holds good with close approxi- 
mation for all dilute solutions under ordinary conditions. 

We shall now consider an isothermal reversible cycle performed 
with a solution of a non-volatile solute in a volatile solvent. Let the 
solution contain n gram-molecules of dissolved substance in W grams 
of solvent, and let the constant absolute temperature of all the processes 
be T. From the solution let there be removed by means of a piston 
and a diaphragm permeable only to the solvent, a quantity of the latter 
which in the solution contained one gram molecule substance dissolved 
in it. This quantity is Wjn grams, and the solution is supposed to be 
present in such large proportions that the removal of this amount of 

1 The proof here given is tali en from an article by Lord Kayleigh, Nature, vol. lv. 
p. 253, 1897. 


solvent does not sensibly affect its concentration, the osmotic pressure 
thus remaining constant during the operation. Since the change in 
volume of the solution is the gram molecule volume, the work done on 
the system in removing the solvent is the product of this into the 
osmotic pressure, and is therefore equal to RT, if the gas laws apply 
to dissolved substances. This quantity of liquid solvent is now con- 
verted into vapour reversibly by expanding at the vapour pressure of 
the liquid, which we shall call /. The vapour pressure of the solution 
is smaller than this and equal to /'. Let the vaporous solvent there- 
fore expand reversibly till its pressure diminishes to this value. The 
gaseous solvent is then in equilibrium with the solution, and may be 
brought into contact with it and condensed reversibly at the pressure 
/', so that the whole system regains its initial state. We have now to 
consider the work involved in the expansion and contraction. The 
work done by the system on expanding from liquid to vapour under 
the pressure /, is equal to the work done on the system in condensing 
the gas to liquid under the constant pressure /', being equal in each 
case to RT for the gram molecule, if we neglect the volume of the 
liquid (cp. p. 29). There remains then the work done by the gas on 
expanding from / to /'. For isothermal expansion we have the work 

RT— per gram molecule of gas (equation la, p. 317). Now 

dp _dv 
p v 

for a gas, since pv = const., and therefore pdv + vdp = 0. For the 
work done during the expansion of the gas then we have - RT-—, or 

- RT per gram molecule if the difference between / and /' is very 

small. For the actual amount of solvent considered we have conse- 

W f-f 
quently - —RT , where M is the molecular weight of the 

solvent in the gaseous state. Since in the whole cycle no heat is con- 
verted into work or vice versa, this work done by the system must be 
numerically equal to the work done on the system during the osmotic 
expulsion of the solvent, i.e. 

W f-f 

ET=~r'RT' J —/- 
Mn f 

f-f Mn 
or - — -- = — 

/ w 

which is identical with the result arrived at for very dilute solutions 
by the method of calculation given on p. 170. Thus by a direct 
thermodynamical proof we can arrive at the relation between the 


osmotic pressure and the lowering of the vapour pressure of liquids by 
substances which are dissolved in them. 

If the solution considered be not very dilute, we cannot write 

for — . For greater concentrations we integrate this expression, and 

thus obtain \og e — ,. By the same process of reasoning as before, we 

then get 

This expression holds good for all concentrations of solutions for which 
the total volume of the liquid does not change when the quantity of 
solvent considered is added to or removed from the solution. If this 
condition is not observed the result is only approximate, for in the 
deduction of the relation we assumed that the work done by the 
expressed liquid in expanding at the constant gaseous pressure of the 
solvent was equal to the work done on the vapour when it was con- 
densed at the constant vapour pressure of the solution, an assumption 
which is only valid if the volume of the liquid is the same before and 
after the reversible mixing of the solvent with the solution. 

The relation between the more accurate logarithmic and the usual 
approximate expression may be obtained by writing the former as 
follows : — 

log.(l + ^). 

If we expand the logarithm in this second form, the first term of the 

expansion is - — /- y identical with the usual expression. 

Having thus obtained the formula for the lowering of the vapour 
pressure of a solvent, we may now proceed to deduce the formula for 
the corresponding rise in the boiling point. From equation (6) we 
obtain the expression 

dP_ Q 

p ~~w 

to represent the concomitant variations of temperature and vapour 

pressure of a solvent. Consider now a solution containing n gram 

molecules of dissolved substance in W grams of solvent. Let this 

solution have the vapour pressure P at the temperature T + dT, T being 

the temperature at which the solvent has the same pressure. At the 

temperature T + dT the solvent will have the pressure P + dP. Now 

p — -Tp is the lowering of the vapour pressure of the solvent, but since 


dP is very small compared with P we may write instead of this the 

expression — for the lowering. But we found above that this lower- 

mg is equal to -==, so that 

^L-J AT 

W ~~2T 2 

Now Q is the latent heat of u gram molecule of the solvent, and M is 
its molecular weight (both molecular quantities referring to the gaseous 
state), so that QjM=q the latent heat of vaporisation per gram. We 
thus obtain 


or dT= •— . 

q W 

T + dT-T=dT is the elevation of the boiling point of the solvent 

caused by the substance dissolved in it, and we have now obtained an 

expression for this in terms of the boiling point of the solvent itself, 

its latent heat of vaporisation and the concentration of the solution, to 

which the elevation is proportional. The elevation of the boiling 

point caused by the solution of one gram molecule of substance in 1 00 

grams of solvent is sometimes referred to as the "molecular elevation'' 

(p. 185). For this concentration n becomes 1 and W becomes 100, 

0'02T 2 
so that the molecular elevation is . For a solution containing 

one gram molecule per 1000 grams, i.e. very nearly one gram molecule 

0-0027 72 
per litre for aqueous solutions, the elevation is — . 

Since both boiling point and heat of vaporisation vary with 
the pressure at which ebullition takes place, there is no definite 
"molecular elevation" for any one solvent unless the pressure is 
specified. For ordinary purposes the pressure is of course the atmo- 
spheric pressure, and the fluctuations to which this is subject have so 
little effect on the molecular elevation that it may be taken as a 
constant magnitude in the practical molecular weight determination 
by the boiling point method. In order to give an example of the 
agreement between the calculated molecular elevation and the same 
magnitude as determined experimentally, we may take the common 
solvent ether. The boiling point of ether is 35°, and therefore T = 
308. The latent heat of vaporisation per gram at this temperature is 

0-022 72 
90. The expression has thus the value 21*1. The average 

molecular elevation observed in the case of nine different substances 


in moderately dilute ethereal solution was found to be 21*3, the 
extreme values being 20*0 and 21*8. 

The expression for the molecular depression of the freezing 
point has a similar form, and may be deduced by means of a 
reversible cycle as follows. Let a solution containing n gram mole- 
cules of dissolved substance in W grams of solvent be contained 
in a cylinder provided with a semi -permeable end, and a movable 
piston. At T - d T, the freezing point of the solution, let such a 
quantity of the solvent freeze out as originally contained one gram 

molecule of substance dissolved in it, viz., — grams. The quantity of 

solution is supposed to be so great that the freezing out of this amount 

of solvent does not appreciably affect its concentration, so that the 

temperature of equilibrium between solution and solid solvent does 

not change during the process of freezing. The solid is now separated 

from the solution, and the whole system is raised to the temperature 

T, the melting point of the solvent, and at this temperature the solid 

solvent is allowed to melt. In doing so it absorbs — q calories, if q 

represents the latent heat of fusion per gram. The fused solvent is 

now brought into contact with the solution through the semipermeable 

diaphragm under equilibrium conditions, viz., with the pressure on the 

solution equal to the osmotic pressure P. By raising the piston under 

this constant osmotic pressure, the solvent passes through the 

diaphragm and mixes reversibly with the solution, the concentration 

as before remaining unchanged. The work done on the piston is 

equal to the product of the constant osmotic pressure P into the 

volume v which contains one gram molecule of solute. But this amount 

of work, according to the osmotic pressure theory, is equal to RT ; 

or if we express the gas constant in thermal units, to 2 T. The system 

after mixing is finally cooled to the original temperature T - d T so as 

to complete the cycle. By selecting the solution sufficiently dilute 

we may make the depression of the freezing point dT as small as we 

choose, and consequently the heat absorbed and evolved in warming 

and cooling the system through this small range of temperature may 

be made negligible in comparison with the finite amount of heat — q 

absorbed by the solvent on melting. In the reversible cycle, then, we 

have the amount — q absorbed at the higher temperature T, and the 

dT W 
amount of this converted into work is — • — q. But the only work 

JL fit 

done by the system is the osmotic work, since the external work 
brought about by the volume-changes on freezing and melting are so 
small as to be negligible. We have consequently 


T n q ~ 2I > 

q W 

The depression of the freezing point is thus seen to be proportional to 

the concentration of the solution, and if we make the concentration 

such that n = 1 and W — 100, that is, if we dissolve one gram molecule 

in 100 grams of solvent, we get the molecular depression equal to 

0-02T 2 

. The expression is exactly the same as that for the elevation 

of the boiling point, q referring here, however, to heat of fusion 
instead of to heat of vaporisation. 

The following table exhibits the nature of the agreement between 
the calculated and observed values of the molecular depression in 
various solvents : — 




Mol. Dep. 



Formic acid 



Acetic acid 












Ethylene dibromide 



Electrical and Chemical Energy 

It is possible, as Helmholtz showed, to establish a relation between 
the energy of the chemical processes occurring in a galvanic cell and 
the electrical energy produced by the action. At one time it was 
supposed that all the energy which could be obtained under ordinary 
circumstances as the heat of the chemical reaction, could be converted 
into electrical energy and made available as an electrical current. 
Thus if Q were the thermal effect for one gram equivalent of the sub- 
stances entering into chemical action in the cell, it was assumed that 
an amount of electrical energy equivalent to this could be obtained 
from the cell for each gram-equivalent of chemical transformation. In 
a Daniell cell, for example, where the reacting system is 

Ou, CuS0 4 solution ; ZnS0 4 solution, Zn, 

the total chemical action is 

Zn + CuS0 4 = Cu + ZnS0 4 , 

zinc being dissolved up at one pole of the battery and copper being 
deposited at the other. Now the thermal effect of this reaction is 
obtained by subtracting the heat of formation of the zinc sulphate 


from that of the copper sulphate, both numbers referring to aqueous 
solutions of the salts (cp. p. 121). We thus get 106,090 - 55,960 = 
50,130 cal. per gram molecule, or 25,065 cal. per gram -equivalent. 
That is, when 32*5 grams of zinc displace an equivalent amount of 
copper from a solution of copper sulphate, 25,065 calories are evolved. 
Now, if the displacement takes place indirectly in a galvanic cell with 
production of electric current, 96,500 coulombs of electricity will be 
obtained, according to Faraday's Law, for every 32*5 grams of zinc 
dissolved. To express the numerical equivalence of thermal and 
electrical energy we have the equation 1 volt-coulomb = 0*24 cal. 
The electrical energy then equivalent to the heat of the chemical 
reaction is 25,065-7-0*24 = 104,270 volt-coulombs. If we now divide 
this number by 96,500, the number of coulombs produced, we get 1*08 
for the electromotive force of the cell expressed in volts, on the 
assumption that all the chemical energy of the reacting substances has 
been converted into electrical energy. Direct measurement of the 
E.M.F. of the Daniell cell gives 1'09 to 1*10 volt. The assumption 
then that the chemical energy is wholly converted into electrical 
energy is in this case very nearly true, and several other cells are 
known to which a similar simple calculation for their electromotive 
force is applicable. These cells, however, are all of a special type,, 
and we shall therefore proceed by means of a reversible cycle of 
operations to deduce a formula of more general application. 

In the first place the cell considered must be, like Daniell's, of a 
completely reversible or non-polarisable type; i.e. it must be of such 
a kind that if we pass a current through it in the reverse direction of 
the current which the cell would of itself generate, the chemical action 
in the cell will be exactly reversed. Thus when the Daniell cell is in 
action the positive current within the cell moves with the positive 
ions from the zinc electrode to the copper electrode. If we now by 
using an external electromotive force make the current pass from the 
copper pole to the zinc pole, the chemical action which occurs is 

Cu + ZnS0 4 = Zn + CuS0 4 , 

which is the reverse of the primary action of the cell, copper being 
dissolved and zinc deposited. 

Let the cell act at the constant temperature T y and generate the 
quantity C = 96,500 coulombs of electricity. If the electromotive 
force of the cell is E, the electrical energy afforded by the cell is EC, 
which may or may not be equal to Q, the diminution in chemical 
energy, which under ordinary circumstances would be the heat of the 
chemical action. Suppose the electrical energy produced to be less 
than the fall in chemical energy, then the element on working will 
give out Q - EC as heat at the constant temperature T. Let now the 
system be heated to the slightly higher temperature T + dT, and let 
the quantity C of electricity be sent through the cell in the reverse 


direction, the temperature being kept constant. If the electromotive 
force has diminished by the amount dE owing to the change of 
temperature, the work done on the cell will be C(E - dE), and the 
amount of heat absorbed will be Q - C(E - dE) on the supposition 
that the heat of the reaction does not change appreciably with the 
temperature. The system is finally cooled to the original temperature 
T, so that everything regains its initial state, a reversible cycle having 
been performed. On the whole, the system has done the external 
electrical work C(E - dE) - CE = - CdE, which must be equal to the 
fraction dTjT of the heat given out at the lower temperature. Now 
the quantity of heat given out is Q - CE, so that the heat transformed 
into work is 


and we have therefore the equation 

-CdE = -±{Q-CE); 


E= G + T Tf 

From this relation we see that in order to ascertain the electromotive 
force of an element from the heat of the chemical action within it, we 
must know the rate of change of the electromotive force with the 
temperature. If the electromotive force does not vary with the 
temperature, i.e. if dE/dTbe zero, then the simple formula 

F- Q 

E ~G 

may be used. The electromotive force of a Daniell cell has a very 
small temperature co-efficient, so in its case the simpler formula gives 
a result closely approximating to the truth. The more accurate 
formula gives in this case a still closer approximation to the observed 
electromotive force, and has been experimentally verified in many 
other instances for which the temperature co-efficient is larger. 
If we write the expression in the form 

we see at once that the electrical energy and the chemical energy of 
the process are equal if dEjdT is zero ; that the electrical energy is 
greater than the chemical energy if the temperature coefficient is 
positive, i.e., if the electromotive force increases with rise of tempera- 
ture ; and that the electrical energy is less than the chemical 
energy if the temperature coefficient is negative, i.e. if the electro- 


motive force falls with rise of temperature. If a cell with a positive 
temperature coefficient of electromotive force is allowed to act, it will 
make up for the difference between the electrical and chemical energies 
by abstracting heat from neighbouring bodies; or, if no external heat 
is available, it will cool itself by working. The student is apt to 
imagine that this is a contradiction of the Second Law of Thermo- 
dynamics ; but, like the self-cooling of a freezing mixture, the process 
here involved is not a cycle of operations, and the system cannot 
regain its original state without work being done upon it. What the 
Second Law contradicts is the existence of a system which, working in 
a cycle, by repeated self-cooling, can convert into work the heat of 
neighbouring bodies. 

An excellent elementary account of the Principles of Thermodynamics 
is that by Carey Foster in Watts' Dictionary of Chemistry, original edition, 
3rd supplement, pt. ii. pp. 1922-1951. 


Accelerating influence of acids, 254, 271 
Acids, dissociation constants, 144, 224, 

electric conductivity, 224 

strength (avidity), 266, 269, 275, 292 
Active mass, 236 

of solids, 243 
Additive properties, 136 
Adiabatic processes, 313 
Affinity constants, 273 
Allotropic transformation, 101 
Amorphous state, 60-62 
Argon, 17, 36, 49 
Association, molecular, 193 
Asymmetric carbon atom, 140 
Atom, 8 
Atomic heat, 31 

hypothesis, 8-21 

refraction, 139 

volume, 42, 137 

weights, 12, 49 

table of, 20 

unit of, 13 

Avidity, 269, 275, 292 

Avogadro's Law, 11 

Balanced Actions, 234-253 
Bases, conductivity, 224 

strength, 276 
Blagden's Law, 63 
Boiling points, 131, 198 

of solutions, 80, 173 
Border curve, 77 
Boyle's Law, 27 

deviations from, 89-91 

for dissolved substances, 163 

Calorie, 6 

Calorimeter, 125 

Capillarity, 189 

Catalysis of ethereal salts, 256, 272 

Circular polarisation, 139, 141, 150 

Colligative properties, 147 

Combining proportions, 9 

Combustion, heat of, 122, 130 
Complex ions, 305, 308 

molecules, 193 
Condensation and vaporisation, 73-83 
Conductivity, molecular, 211, 220-224 
Conservation of energy, 4, 119, 311 
Constant boiling mixtures, 81 
Constitution and physical properties, 136- 

Continuity of gaseous and liquid states, 

77, 91 
Corresponding conditions, 94 
Critical constants, 73 

solution temperature, 76 
Cryohydrates, 65 
Crystalline liquids, 62, 104 
Crystallisation, 61 
Cycle of operations, 314 

Dalton's law of partial pressures, 55, 79 
Decomposition by water (hydrolysis), 247, 

Degree of dissociation, 221, 231, 273 
Dehydration of hydrates, 111, 244 
Deliquescence, 114 
Density, 3, 127 

of gases, 13, 176 

of solutions, 155 
Depression of freezing point, 174, 186, 

Desmotropy, 253 
Diffusion of gases, 148 

in liquids, 149, 165 
Dilution formulae, 224, 227 
Diminution of solubility, 293, 302 
Dissociation constants, 144, 224, 285 

electrolytic, 217-233 

gaseous, 241, 265 

pressure, 244, 250 
Distillation of mixtures, 80 

with steam, 82 
Double decomposition, 305 
Dulong and Petit's Law, 30 
Dynamic isomerism, 253, 265 


Efflorescence, 113 
Electrical energy, 7, 329 

units, 6 
Electrolysis, 201-211 
Electrolytes, 202 

Electrolytic conductivity, 201, 207, 211, 
272, 277 

dissociation, 217-233, 272, 296-310 
Electromotive force, 6, 330 
Elements, 8 

periodic classification, 44 

table, 20 
Elevation of boiling point, 173, 181, 327 
Endothermic compounds, 123 
Energy, 4 

conservation of, 4, 119, 311 

intrinsic, 118 
Equations, chemical, 22-26 

therniochemical, 120 
Equilibrium, 97-116 

chemical, 234-253 

of electrolytes, 283-295 
Equivalent weights, 9 

determination of, 17 
Eutectic mixtures, 71 
Exothermic compounds, 123 
Explosions, 262 
Extraction with ether, etc., 57 

Faraday's Law, 204 
Freezing point, 61 

depression of, 174, 186, 329 

of mixtures, 70 

of solutions, 63, 174 
Fusion and solidification, 60-72 

Gas-constant, 29 
Gas-laws, 27-29 

deviations from, 89-91 

for solutions, 162-164 
Gaseous diffusion, 148 

liquefaction of, 74 
Gases, solvent action of, 79 
Gay-Lussac's Law of Volumes, 11 

of expansion, 27 

Heat, atomic, 31 

mechanical equivalent of, 6 

molecular, 33, 35 

specific, 30-37 

units, 6 

of combustion, 122, 130 

of formation, 121 

of transformation, 117 
Henry's Law, 55, 198, 321 
Homologous series, 127-135 
Hydrates, 51, 67, 69, 108 

dehydration o£ 111, 244 
Hydrolysis of ethereal salts, 238, 256 

of salts, 247, 278 

Indicators, 309 

Intrinsic energy, 118 

Inversion of cane-sugar, 254, 271 

Inversion points, 102 

Ions, 204 

migration of, 207 

speed of, 212 
Isohydric solutions, 286 
Isomerism, 145, 146 

dynamic, 253, 265 
Isothermal curves, 76, 91 
Isotonic solutions, 167 

Jellies, 213 

Kinetic theory, 84-96 

Liquefaction of gases, 74 
Liquids, molecular weight, 188 

associated, 53, 195, 200 

crystalline, 62, 104 

normal, 53, 188, 199 
Lowering of vapour-pressure, 171, 325 

of freezing point, 174, 328 

Magnetic rotation, 141 
Mass, 2 

active, 236 
Mechanical equivalent of heat, 6 
Medium, influence on rate of chemical 

change, 261 
Melting points, 60, 69, 72 

in homologous series, 133 
Metastable conditions, 70, 101, 115 
Migration of ions, 207 
Miscibility of liquids, 52 
Mixtures, constant-boiling, 81 

distillation of, 80 

eutectic, 71 

of liquids, 52 

of gases, 55 
Moduli, 155 
Molecular complexity, 193-200 

conductivity, ,211, 220-224 

depression, 187, 329 

elevation, 185, 327 

heat, 33, '34, 128, 137 

magnetic rotation, 142 

refractive power, 139 

rotation, 139 

volume, 14, 128 

weights, 14, 176-192, 193-200 

Neumann's Law, 33 
Neutralisation, heat of, 119, 296 
Normal liquids, 53, 188, 199 

Octaves, law of, 42 
Ohm's law, 217 
Optical activity, 139, 150 
magnetic, 141 



Osmotic pressure, 158-168, 324 
Ostwald's dilution formula, 224 
Oudeman's Law, 153 
Oxidation in solution, 300 

Partial pressure, 55, 79, 159 
Partition coefficient, 59, 197 
Periodic law, 38-49 

table, 44 
Phases, 97 

new, 114 
Physical properties and chemical constitu- 
tion, 142-147 
Precipitation, 303-304 
Pressure, atmospheric, 3 

osmotic, 158-168, 324 

partial, 55, 79, 159 
Prout's hypothesis, 21 

Bate of chemical action, 254-265 

of crystallisation, 61 
Reactions of salts, 299 
Recrystallisation, 57 
Reduction in solution, 300 
Refractive power, 138 
Reversible cycles, 314 

processes, 315, 321 
Rotatory power, 139 

magnetic, 142 
Rudolphi's dilution formula, 227 

Salting-out, 303 

Salts, acid, 285 

Saponification of ethereal salts, 256 

Saturated solutions, 50 

vapour, 75 
Semipermeable membranes, 160 
Solid solutions, 191 
Solidification and fusion, 60-72 
Solubility, 50-59 

curves, 51, 107 

of electrolytes, 293 

of gases, 55, 59 

of "insoluble" sajits, 307 

of isomers, 135 

of precipitates, 306 

in homologous series, 134 
Solutions, boiling point, 80 

freezing point, 63 

isotonic, 167 

vapour pressure, 79 
Solution-tension, 96 
Solvent action of gases, 79 

Specific gravity, 3, 127 

heats, 30-39 

refractive power, 138 

magnetic rotation, 142 

rotatory power, 139 

volume, 3, 127 
Speed of gas-molecules, 87 

of ions, 212 
Strength of acids and bases, 266-282 
Sublimation, 78 
Sugar-inversion, 254, 271 
Superfusion, 61, 62 
Supersaturated solutions, 50, 115 
Surface tension, 188 

Tautomerism, 253 
Temperature, 5 

influence on molecular conductivity, 222 
rate of chemical change, 260 

of ignition, 262 
Thermochemical change, 117-126 
Thermodynamics, 311-332 
Transition points, 102 
Transport numbers, 210 
Triple point for water, 99, 321 
Trouton's rule, 124 

Units, 1-7 

for atomic weights, 12 

Valency, 9, 46 

Valson's moduli, 155 

Van der Waals's equation, 89-96 

Van 't Hoff's factor *, 229 

dilution formula, 227 
Vaporisation and condensation, 73 - 83, 

Vapour density, 176 

pressure, 75, 98 
of solids, 78 

of solutions, 79, 171, 325 
Velocity, constant, 237 

of chemical action, 254-265 
Volume, atomic, 42, 137 

critical, 73 

molecular, 14, 128 

specific, 3 

Water, decomposition by, 247, 278 

influence of vapour, 265 
Welter's rule, 122 

Work done by an expanding gas, 29 
Wiillner's Law, 79 


Printed by R. & R. Clark, Limited, Edinburgh 



£~Zk f^&^ia^ ^ 


I '^Z^^^^gg 






-■rut OF 25 CENTS 
*>*ANC*»ITU|- FI NE d „.r uRE to return 


•thVs book oh™"*^ on the fourth 


DAY AND TO *l.w 


AUG 30 19CL 
ftOV 6 1936 




FEB 10 1940 
JUN 7 1941 


LD 2l-20wi-6,'33